Distribution of Zeros of Entire Functions (Translations of Mathematical Monographs)

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VOLUME

5

B.Ja. Levin

Distribution of Zeros of Entire Functions Revised Edition

American Mathematical Society

Translations

of Mathematical Monographs

Distribution

Volume 5

of

ZEROS OF ENTIRE FUNCTIONS Revised Edition

by B.

Ja. Levin

AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND

PACTIPEJlEJIEHHE KOPHEA UEJIhIX YHKUHA E. H. JIEBHH rocy.napcrBeHHoe H3J(8Te.jILCTBO

TexHHI
MOCKBa 1956

Translated from the Russian by R. P. Boas, J. M. Danskin, F. M. Goodspeed, J. Korevaar, A. L. Shields, H. P. Thielman

Publication aided by grant NSF-GN 57 from the NATIONAL SCIENCE FOUNDATION

Library of Congress Cataloging in Publication Data

"'"

Levin, Boris IAkov1evich. Distribution of zeros of entire functions. (Translations of mathematical mono~r~hs ; v. 5) Translation of Rasprede1enie korne1 tselykh

fllIlkfS il'.

Bibliography: p. 1. Functions, Entire. I. Title. II. Series. QA353.E5L4813 1980 515.9'8 80-36891 ISBN 0-8218-4505-5 Copyright © 1964, 1980 by the American Mathematical Society Second printing, f"ust edition, 1972

All rights reserved. No portion of this book may be reproduced without the written permission of the publisher. Printed in the United States of America

TABLE OF CONTENTS PREFACE TO THE SECOND EDITION . . . . . . . . . . . . . . . . . . . . . . .

ix

PrulFACE TO THE FIRST EDITION. . . . . . . . . . . . . . . . . . . . .

xi

CHAPTER

I

THE GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS . . • . . • . . . . . . . • . . . . . . . . . . . . . . . . .

1

1. The growth scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. The connection between the growth of an entire function and the rate of decrease of its Taylor coefficients. . . . . . . . . 4 3. The expansion of entire functions in infinite products. . . . . . . 6 4. Estimates for canonical products . . . . . . . . . . . . . . . . . . . . 9 5. Jensen's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 6. The relation between the maximum modulus of ~ holomorphic function and the maximum of its real part . . . . . . . . . . . . . 17 7. Lower bounds for the modulus of a polynomial . . . . . . . . . . 19 8. Lower bounds for the modulus of a holomorphic function . . . . 21 9. The growth of the product of two entire functions . . . . .. . 22 10. Hadamard's theorem. . . . . . . . . . . . . . .. . 24 11. Entire functions of integral order. . . . . . . . . . . . . . . .. . 27 12. Proximate orders. . . . . . . . . . . . . . . . . . . . . . . . . .. . 31 13. Extension of the classical theorems to proximate orders . . . . . 41 14. The Phragmim-LindelOfprinciple . . . . . . . . . . . . . . .. . 47 15. The indicator function . . . . . . . . . . . . . . . . . . . . . .. . 51 16. The fundamental relations and analytic properties of the indicator function . . . . . . . . . . . . . . . . . . . . . . . . . . .53 17. Auxiliary functions . . . . . . . . . . . . . . . 61 18. The generalized indicator. . . . . . . . . . . .70 19. Plane convex sets . . . . . . . . . . . 74 20. Entire functions of exponential type .... . . 84 CHAPTER

II

ENTIRE FUNCTIONS WHOSE SETS OF ZEROS HAVE AN ANGULAR DENSITY. . . . . • . • . . . . . . . . . . . . . • . . .

89

l. Basic results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 2. Entire functions of nonintegral order with regular distribution of zeros (proof of theorem 1). . . . . . . . . . . . . . . . . . . . . . 97 3. Entire functions of integral order with regular distribution of zeros (proof of theorem 2) . . . . . . . . . . . . . . . . . . . . . . . 108 v

vi

TABLE OF CONTENTS

4. Construction of an entire function with a given indicator (proof of theorem 3) . . . . . . . . . . . . . . . . . . . . . . . .. 5. Asymptotic representation of entire functions with regular distribution of zeros (proof of theorem 4) . . . . . . . . . . . . . 6. Entire functions with regular sets of zeros (proof of theorem 5) . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Theorems on equicontinuity (proof of theorems 6 and 7) ...

III

116 122 125 128

FUNCTIONS OF COMPLETELY REGULAR GROWTH . . • • . .

139

1. Set ofrays of completely regular growth . . . . . . . . . . . . . . 2. A generalized formula of Jensen. Investigation of the function JF(O) • • . . . . . . • . . . . . . . . . . . . . . . . • • . . • 3. The basic theorem on functions of completely regular order.. 4. Indicator of the product of two functions . . . . . . . . . . . . . 5. Some consequences of the generalized formula of Jensen. Case ofa sinusoidal indicator. . . . . . . . . . . . . . . . . . . . .

140

CHAPTER

CHAPTER

IV

UNIQUENESS, INTERPOLATION AND COMPLETENESS ••••

142 152 159 161 168

1. Uniqueness theorems for entire functions of finite order . . . . 170 2. Uniqueness theorems for functions of finite order, holomorphic in an angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 3. Functions vanishing on a set with an angular density . . . . . 190 4. Representation of entire functions of Lagrange interpolation series . . . . . . . . . . . . .~ . . . . . . . . . . . . . . . . . . . . . . . 194 5. Some applications of Lagrange interpolation series . . . . . . . 202 6. Completeness of systems of functions. The connection between completeness and uniqueness . . . . . . . . . . . . . . . 210 7. Theorems on the completeness of some systems of entire functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 CHAPTER

V

FUNCTIONS OF CLASS

A . . . . . . . . . . . . . . . . . . . . . 222

1. Formula of Carleman. Criterion for an entire function of exponential type to be of class A ........... . . . . . . . 2. Representation of a function harmonic in a half-plane . . . . . 3. Representation of a function of exponential type and of class A in the upper half-plane . . . . . . . . . . . . . . . . . . . . 4. Functions of class A and of completely regular growth . . . . . 5. Indicator diagram of an entire function of exponential type, and of class A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Theorem of M. G. Krein on the expansion of the reciprocal of an entire function. . . . . . . . . . . . . . . . . . . . . . . . . . . .

223 230 236 243 252 258

vii

TABLE OF CONTENTS CHAPTER

VI

ROOTS OF EXPONENTIAL SUMS. . .

.264

1. Information from the theory of almost-periodic functions ... 2. Roots of an almost-periodic function with a bounded spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. A general theorem on roots and on me~n motion for holomorphic almost-periodic functions. . . . . . . . . . . 4. A theorem on mean motion for an almost-periodic function with semi bounded spectrum . . . . . . . . . . . . . . . . . . . . . 5. Functions approximated by exponential polynomials. . . . . . 6. Growth of a function of class EJ for a norming region in the form of a polygon. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Growth of a function of class E I outside an arbitrary norming region I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER

VII

264 268 273 279 287 294 301

THE THEOREM OF HERMITE-BIEHLER FOR ENTIRE FUNCTIONS . . . . . . . . . . . . . • . . . . . . . . . . . . . •

305

1. Representation of a real merom orphic function mapping the upper half-plane onto the upper half-plane . . . . . . . . . . . . 307 2. Generalization of the theorem of Hermite- Biehler to arbitrary entire functions. . . . . . . . . . . . . . . . . . . . . . . . . 311 3. Representation of a function of class H B . . . . . . . . . . . . . 317 4. The Hermite-Biehler theorem for entire functions of exponential type . . . . . . . . . . . . . . . . . . . . . . . . .. 318 CHAPTER

VIII

APPROXIMATION OF ENTIRE FUNCTIONS BY POLYNOMIALS WHOSE ZEROS LIE IN A GIVEN REGION . . . . •

327

1. Functions which can be approximated by polynomials, all of whose zeros lie in an angle. . . . . . . . . . . . . . . . . . . 327 2. Theorems on composition of polynomials . . . . . . . . . 336 . 340 3. Multiplier sequences. . . . . . . . . . . . . . . . . . . . . CHAPTER

1. 2. 3. 4. 5. 6. 7.

IX

OPERATORS PRESERVING INEQUALITIES AMONG ENTIRE FUNCTIONS • . . . . . . . . . • • . . • • . . • . • • . • . . • .

348

Majorants and admissible classes . . . . . . . . . . . . . . . . . . Some properties of the class P*. . . . . . . . . . . . . . . . . . . . Operators preserving subordination (58 r operators) . . . . . . . The class P and inequalities on the real axis . . . . . . . . . . . Classes of functions of several variables . . . . . . . . . . . . . . General form of the operators 58 and 58 * ............. Certain extremal properties of entire functions. . . . . . . . . .

350 352 355 362 366 373 380

viii

TABLE OF CONTENTS .

383

The impossibility of constructing an exact scale of growth . Convergent and divergent types . . . . . . . . . . . . . . . . . . . The Paley-Wiener Theorem . . . . . . . . . . . . . . . . . . . . . Levitan polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . Power series with coefficients that are the values of entire functions of exponential type for integral values of their arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

383 384 386 391

APPENDIX

1. 2. 3. 4. 5.

APPENDIX

I

SOME ADDITIONAL PROBLEMS OF THE GENERAL THEORY

II

393

THE APPLICATION OF UNIQUENESS THEOREMS TO QUASIANALYTIC CLASSES OF FUNCTIONS . . . . • . . . . . •

395

1. Functions defined by their values on an interval. . . . . 395 2. Quasianalytic classes of almost-periodic functions . . . . . . . . 405 APPENDIX

III

COMPLETENESS AND LINEAR INDEPENDENCE ON A FINITE INTERVAL OF SYSTEMS OF FUNCTIONS

IeiAP I ...

412

1. Completeness theorems . . . . . . . . . . . . . . . . . . . . . . . . 412 2. Minimality (strong linear independence) of the system IeiAk I 419 %

APPENDIX

IV

ApPLICATION OF UNIQUENESS THEOREMS TO SOME PROBLEMS IN THE THEORY OF DIFFERENTIAL EQUATIONS . . . . . . . . . . . . . . . . . . . . . . . • . • . .

424

1. Completeness of a system of solutions of a differential equation of second order in regions in the complex plane . . . . 424 2. The inverse problem for the Sturm-Liouville equation . . . . . 430 APPENDIX

V

REPRESENTATION OF A POSITIVE ENTIRE FUNCTION OF EXPONENTIAL TYPE AS THE SQUARE OF THE ABSOLUTE

APPENDIX

VALUE OF AN ENTIRE FUNCTION • . . . . • . . . • • . • . . .

437

ALMOST-PERIODIC FUNCTIONS WITH BOUNDED SPECTRA.

444

VI

1. An expression for the real part of a zero of a function of the class [.:1J .................................. 444 2. A characterization of the set of zeros of an almost-periodic function of the class [.:1 J . . . . . . . . . . . . . . . . . . . : . . . . 449 3. The relation between the Fourier series of 1/t(x) and f(x) . . . . 459 APPENDIX

VII

MISCELLANEOUS THEOREMS AND PROBLEMS . • . • . • .

464

APPENDIX

VIII

FURTHER DEVELOPMENT OF 11iE THEORY OF FUNCTIONS OF COMPLETELY REGULAR GROWTH • . • • ••

470

LIST OF IMPORTANT IDEAS AND THEOREMS . . . . . . . • . • . . • . . • . . 492 BIBLIOGRAPHY

•.••••••.••••••.....•..•.••••••.•.

508

PREFACE TO THE SECOND EDmON

In this edition I have corrected some inaccuracies and misprints, of which there were a rather large number in the first edition. Since the appearance of the first edition many papers have been published on the theory of functions of completely regular growth and related topics. However, I do not have the time to expound these new results, which would fill an additional volume. I have tried to make up for this shortcoming by stating new results in a number of places, without proofs, when they are connected with the topics under discussion. I have also added an appendix, "Further development of the theory of functions of completely regular growth," in which I present a survey of recent results. In addition, I have cited papers in which this material is presented in more detail, with proofs. B. Ja. Levin 1978

ix

PREFACE TO THE FIRST EDmON

One of the most important problems in the theory of entire functions is the problem of the connection between the growth of an entire function and the distribution of its zeros. Many other problems in fields close to complex function theory lead to this problem. The connection between the growth of an entire function and the distribution of its zeros was investigated in the classical works of Borel, Hadamard, Lindelof and other authors at the end of the last and the beginning of the present century. More precise characterizations of the growth and of the distribution of the zeros of an entire function led to the possibility of establishing more precise relations between them. At the same time analogous result'; were obtained for a wider class of functions, holomorphic in an angle. Especially precise relations were obtained for a special class of functions which it is natural to call the functions of completely regular growth. In this book the theory of functions of completely regular growth is applied systematically to the investigation of various questions in the theory of entire functions. Chapters II and III are devoted to the theory of functions of completely regular growth. Then in Chapters IV, V, and VI this theory is applied to various questions of completeness and uniqueness, interpolation, distribution of the roots of exponential sums, properties of entire functions bounded on the real axis, and so forth. In the succeeding chapters we study a range of questions connected with carrying over to entire functions certain properties of polynomials. This includes the generalization of the Hermite-Biehler theorem, begun in the works of N. G. Cebotarev and developed in the works of other Soviet mathematicians (Chapter VII), a presentation of the results of Laguerre-Polya and Schur on algebraic properties of entire functions (Chapter VIII) and a far-reaching generalization of the inequalities of A. A. and V. A. Markov and of S. N. Bernstein (Chapter IX). In these chapters the machinery of the theory of functions of completely ~ular growth does not playa decisive role; however, it simplifies the investigation and shortens some of the proofs . .Since I was unable to find a suitable reference where the general theory of entire .functions is presented in the generality and in the form that I require, I felt It necessary to expand the introduction I had originally planned into a .~ extensive chapter devoted to the general theory of entire functions. xi

xii

PREFACE

Chapter V can be read independently of Chapter IV, and VII, VIII and IX can be read independently of Chapters IV and VI. At the end of the book there are several supplements in which applications of the results and methods presented in the book are given to related fields. The range of questions treated in this book was determined by the iaterests of the author and the direction of his own investigations. At the end of the book there is a list of the most important theorems and concepts. By using this list the reader can quickly recall not only the definitions of the concepts, but also the basic facts relating to them. The classical concepts and theorems are not included in this list. I wish to thank M. G. Krein, who persuaded me to write this book, and who read it in manuscript and made a number of important comments. I also wish to thank N. I. Ahiezer, who read the manuscript as it was being written and made a number of suggestions that I followed. I am also very thankful to N. S. Landkoff who read the entire manuscript very carefully and made a number of comments enabling me to improve the book.

CHAPTER

I

THE GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS In this chapter we present the elements of the general theory of growth of entire functions. An entire function is a function of a complex variable holomorphic in the entire plane and consequently represented by an everywhere convergent power series (1.00) These functions form a natural generalization of the polynomials, and are close to polynomials in their properties. The theorem of Weierstrass on the expansion of entire functions into infinite products provided the basic apparatus for the investigation of the properties of entire functions and was the starting point for their classification. At approximately the same time as this work of Weierstrass, Laguerre studied the connection between entire functions and polynomials, and introduced the important concept of the genus of an entire Cunction. The classical investigations of Borel, Hadamard and Lindelof dealt with the connection between the growth of an entire function and the distribution of its zeros. The rate of growth of a polynomial as the independent variable goes to infinity is determined, of course, by its degree. On the other hand, the number of roots of a polynomial is equal to its degree. Thus, the more roots a polynomial has, the greater is its growth. This connection between the set of zeros of the function and its growth can be generalized to arbitrary entire functions. The content of most of the classical theorems of the theory of entire functions consists in establishing relations between the distribution of the roots of an entire function and its asymptotic behavior as z -+ 00. To measure the growth of an entire function and the density of its zeros, a tpccial growth scale is introduced. 1. 1be Growth ScaIe

To characterize the growth of an entire function fez) we introduce the function . M,(r)= maxlf(z}/

,_,-r

2

GF.NERAL THEORY OF THE GROWTH OF ENTIIO: FUNCTIONS

[CHAP. I

(sometimes we shall simply write M(r». It is easily seen that Mjr) is continuous. In fact, let r l < r2 and If(r2e ifo)1 = M(rJ. Then

o < M(r2 )

-

M(r l ) <; IJ(r2 e ifo)I -

lJ(r1eifo)1 < e

for r 2 - r l < Be. It follows from the maximum modulus principle that as r increases Mjr) grows monotonically. The rate of growth of the function Mjr) is an important characteristic of the behavior of the entire function. Let us show that for an entire function that is not a polynomial Mjr) grows faster than any positive power of r. THEOREM

If there exists a posit ire integer II such

I.

.

Mf(r)

Itm ~
then f(z) is a polynomial of degree at most PROOF.

that

I

II.

If

!(z)=au+a1z+ ... +anz"+an+lZ'Hl+ ... and then the function cp (z)

=

P,,(z»)z-n-l

U(z) -

is entire and tends uniformly to zero on some sequence of circles Izl (r n --+ (0). It follows from the maximum principle that
! (z)

=P

lI

= rn

(z).

Thus, in order to estimate the growth of transcendental entire functions, one must choose comparison functions that grow more rapidly then powers of r. We choose comparison functions of the form

er \

where k > O. An entire functionf(z) is said to be afunction offillite order if there exists Ii positive constant k such that the inequality M,(r)

< e,k

is valid for all sufficiently large values of r (r > ro(k». The greatest lower b<'und of such numbers k is called the order of the entire function f(z). 1

Here, as usual, lim cp (r) r-+oo

=

lim (sup cp (t) ) r-+oo

t>r

and

lim cp (r) r-+<:o

=

lim (inf cp (I) ). r-+oo

t~r

Sec. I)

3

THE GROWTH SCALE

It follows from this definition that if p is the order of the entire functionf(z), and if Ii is an arbitrary positive number, then (1.01) where the inequality on the right is satisfied for all· sufficiently large values of r, and the inequality on the left holds for some sequence {r n} of values of r, tending to infinity. It is easy to verify that condition (1.01) is equivalent to the equation -.- In In M, (r) (1.02) p= r+oo 11m 1n r ' which could therefore be taken as the definition of the order of the function. An inequality that holds for a/l sufficiently large values of r will be called an

asymptotic inequality. For functions of a given order a more precise characterization of the growth is given by the type of the function. By the type (1 of an entire function fez) of order p we mean the greatest lower bound of positive nUIl'.bers A for which asymptotically

M,(r)

< eA,I.

Just as with the definition of order it is easy to verify that the type (1 of a function

fez) of order p is given by the equation '0

=

lim In M,(r) • r-too rP

If (1 = 0, the function fez) is said to be of minimal type, if 0 < (1 < 00 of = 00 of maximal type. An example of an entire function of order n and type (1 is the function

normal type, and if (1

for integral n. It is easy to verify also that sin z is an entire function of order one and normal type The function

(1

=

I. The function cos Vz has order p

= 1 and type C1 =

I.

er

.. an example of an entire function of infinite order. In the following section we construct examples of entire functions of arbitrary order and type. We shaU say that the function h(z) is of larger growth than the function /t(z) if the order of f'l.(z) is greater than the order of Il(z), or if the orders are equal and the type of 12(z) is larger than the type of fl(z). ~\' It is easy to see tliat the order of the sum of two functions is not greater .... the larger of the orders of the two summands, and if the orders of the

4

GENERAL THF.ORY OF THE GROWTH OF ENTIRE FUNCTIONS

[CHAP. I

summands and of the sum are all equal, then the type of the sum is not greater than the larger of the types of the two summands. In addition, if one of the two functions is of larger growth than the other, then the sum has the same order and type as the function of larger growth. 2. The Connection Between the Growth of an Entire Function and the Rate of Decrease of its Taylor Coefficients Let the entire functionf(z) have the power series

(1.03) The radius of convergence is infinite and therefore n _

lim

n-+oo

VI cnl = o.

( 1.04)

The following theorem enables one to determine the o~der and type of an entire function by the rate of decrease of its Taylor coefficients. THEOREM 2. The order and type of an entire function are expressed in terms of its Taylor coefficients by the following equations

p=

(aep)~ = PROOF.

-1.-

nlnn

1m --1-' n-+oo I n -

(1.05)

I enl

liin n-+oo

(n ~VI cn I).

( 1.06)

By a well-known inequality for the coefficients of a power series M,(r)

I cn I -< --,:;a .

(1.07)

If f(z) is of finite order, then asymptotically M,(r)

< eArk ,

(1.08)

Icn 1< eArkr-n. Employing the usual method for finding extrema it is easy to see that the function on the right side of this inequality takes its largest value in the range r > 0 for

and therefore asymptotically

(1.09)

Sec. 2]

TAYLOR COEFFICIENTS

5

Conversely, assume that (1.09) holds for all indices n greater than no(k, A), and let us estimate M,{r). For n > mr = [2 k eAkrk] and all sufficiently large values of r we have by (1.09) and therefore

mr

If(z) I

< ~ Ic"lr"+2- mr • ,,:0

Introducing the notation Il-(r) = max I c" Ir",

"

we have

-< (1 + 2keAkrk) IJ. (r)+ 2- tnr.

(1.10) If j(z) is not a polynomial, then M,(r), and therefore by (\.10) also /J(r), grows faster than any power of r, and therefore the index of the largest term in the series (1.03) increases without bound as r grows. It follows from (1.09) that asymptotically M,(r)

The maximum of the right side is attained for n = Akr k

and therefore asymptotically

-<

... (r) eAr". From (\.10) we see that asymptotically M,(r)

< (2 + 2keAkrk) eAr".

(1.11)

Thus (1.09) follows from (1.08), and (1.11) follows from (1.09). This shows that the order p of an entire function j (z) is equal to the greatest lower bound of numbers k for which the asymptotic inequality (1.09) holds, and the type is equal to the greatest lower bound of numbers A for which the asymptotic inequality (1.09) holds with k = p. From this both assertions of the theorem follow at once. We note an interesting corollary that follows at once from formulas (1.07) and (1.10). The maximal term ,,(r) in the power series expansion of an entire function of finite order satisfies the asymptotic inequalities Il- (r)

-< M (r) < rpHIl- (r),

Where p is the order of the functionj(z) and £ is an arbitrary positive number. With the aid of Theorem 2 one can easily construct entire functions of arbitrary order and type. To this end consider the entire function (X)

f(z)= ~

(Aaz )" fI~ r(an + 1) ,

6

GENERAL THWRY OF THE GROWTH OF ENTIRE FUNCTIONS

where A

> 0 and > O. (l

r(czn

[CHAP. I

By Stirling's formula

+ 1) = (Q;)"n V21tczn [1 +0 (!)]

we have! from (1.05) and (1.06) I

p=anda=A. Q

Functions of maximal and minimal type can also be constructed. For example, for n n)T n

In cn = (

(n

=

1, 2, 3, ... )

the series (1.03) represents a function of order p and maximal type, while for n

en- ( n In1 n )P

(n

= 2,

3, ... )

it represents a function of order p and minimal type. For c" = r"a we obtain an entire function of order zero. All these assertions follow at once from (1.05) and (1.06). Similarly one can construct entire functions of infinite order. The limit superiors in formulas (1.05) and (1.06) are unchanged if e.. is replaced by (n + 1)£,,,. Consequently, order and type are unchanged by differentiation.

3. The ExpansioD of EDtire FuoctioDS iD IDfiDite Products It is well known that every polynomial can be written as the product of linear factors. The analogue of this assertion for entire functions is the theorem of Weierstrass on the representation of entire functions by infinite products. This representation is the basis for the investigation of the basic question of the theory of entire functions-the question of the relation between the growth of the entire function and the distribution of its zeros in the complex plane. Let ai' az, ... , a.. , ... be a sequence of complex numbers, none of which is zero, with the point at infinity as the only limit point. We shall construct an entire function whose set of zeros is precisely this sequence. I

Translator's note: The entire function f(z) =

has order p and type

tI.

aep)",p z·

1: (-;;-

Sec. 3]

EXPANSION OF ENTIRE FUNCTIONS IN INFINITE PRODUCTS

7

We can assume that these points have been arranged in order of increasing moduli (if several different points an have the same modulus, then we take them in any order). Choose a sequence of natural numbers P.. such that the series

~I :" rn+1

(1.12)

n~1

converges uniformly in each bounded domain. Such a choice is possible since, for Izl <; R (R > 0) the inequality

is satisfied for all sufficiently large values of n, and thus, for example, one can choose P.. = n. We form the infinite product co

ll(z) =

II 0(: ; Pn). n=1

(1.13)

II

in which u2

uP

u+-+ ... + O(u; p)=(I-u)e 2 P ,

O(u; 0)= l-u.

The functions G(u; p) are called primary factors. Let us show that the product (1.13) converges uniformly on each closed bOunded set that contains none of the points aft' and therefore defines an entire function that vanishes at the points aft and only at them. To this end we estimate die quantity lin G(u;p)1 for lui <; q < 1 and larg (I - u)1 <; TT. From the .pansion up + 1 up +la In O(u; p)=- p+ 1 - p+2 - ... , ~ich is valid under the conditions indicated, we obtain

lIn

.

a (u,

p) I

< 1I u_IIP+l u I -<

1

It follows from this inequality that for Izi <; Rand n )

Z

an

.

> n(q, R) we will have

IIn 0 (an; Pn 1< T -1 q I 1,,,+1 Z

_!

p+1

1 _ q Iu I

'

~ this implies, because of the uniform convergence of the series (1.12) in the

Ircle Izi <; R, that the series

co

~ n=1

In O(~

; Pn),

also the p~oduct (1.13), con~erges uniformly in each closed subset ';Ibis':he~fore Circle that contams none of the pomts a". fId.· ..

8

GENERAL THEORY OF THE GROWTH OF ENTIRE fUNCTIONS THEOREM

3.

[CHAP. I

Every entirefunctionf(z) can be represented in the form

!(z) =

.

z"'e91.)

II 0(: ; Pn) n=1

(1.14)

"

where g(z) is all entire function, an are the nonzero roots of fez), and m is the order of the zero of fez) at the origin. PROOF.

The nonzero roots a" of the entire function fez) (n = 1,2, ... ,

w ~ 00) have no finite limit point. We form the product

...

II (z) =

p. a (:,. ; p,,),

in which P.. = 0, if w is finite. The function
-

f(z) ztnJI (z)

is entire and has no zeros. Hence the function g(z) = In !fI(z)

is also entire and we obtain the desired representation

! (z) = z'"e' I·) II (z). In the representation (1.14) the sequence of numbers p" is not uniquely determined, and therefore the function g(z) is not determined uniquely either. The representation of the functionf(z) is considerably simpler if the numbers a.. satisfy the following supplementary condition: the series 00

~ \a1n \).

(1.12')

n"l

converges for some positive A. In this case let p denote the smallest integer for which the series

converges. Clearly, 0 ~ p < A. The series (1.12) will converge uniformly if we put allp" = p. The uniformly convergent infinite product 00

I1(z) =

II 0(:" ; p)

(1.15)

n=1

is called a canonical product, and the number p is called the genus of the canonical product.

Sec. 4]

ESTIMATES FOR CANONICAL PRODUCTS

9

In the case just considered it is customary in the representation (1.14) to choose the canonical product for the infinite product. Then the function g(z) is uniquely determined. If g(z) is a polynomial,f(z) is said to be an entire function offinite genus. The larger of the numbers p and q, where q is the degree of the polynomial g(z), is called the genus of the entire functionf(z). If g(z) is not a polynomial, or if the series (1.12') diverges for all values of A, then the genus is said to be infinite.

4. Estimates for Canonical Products The representation of an entire function as an infinite product makes it possible to establish a very important dependence between the growth of the function and the density of distribution of its zeros. As a measure of the density of the sequence of points a", having no finite limit point, we introduce the convergence exponent. By the convergence exponent of the sequence

we mean the greatest lower bound of numbers A for which the series

( 1.16) converges. Clearly the more rapidly the sequence of numbers la,,1 increases, the smaller will be the convergence exponent. In particular, the convergence exponent can be zero. If the series (1.16) diverges for all A > 0, then the convergence exponent is ~d to be infinite. If A is equal to the convergence exponent, the series (1.16) mayor may not converge, depending on the sequence {a,,}. • For example, the sequences a" = nJ/ p and a" = (n lnz n)l/p have the same ijonvergence exponent P, but in the first case the series (1.16) diverges when A = p, while in the second case it converges. The sequences {eft} and {In n} have convergence exponents 0 and 00, respectively. Let us recall again the following obvious relation between the convergence exponent PI of the sequence {a .. } and the genus p of the corresponding canonical product (US)

-;h

P
is an integer, then p = PI when the series (1.16) diverges for l = PI' while + I means that the series converges. '_ A more precise description of the density of the sequence {a .. } than the conIIIpnc:e exponent is given by the growth of the function nCr) equal to the

~ .. p

10

GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

number of points of the sequence in the circle monotone function we mean the number

_

Izl

< r.

By the order of this

11m In n (r)

Pl- r-+co

[CHAP. I

(1.17)

Inr'

and by the upper density of the sequence {an} we mean the number ...

Ll=

1~ n(r) 1m - - . r P'

(1.18)

r .... oo

If the limit exists, then A is simply called the density. LEMMA

I.

The converKence exponent of the sequence

a1 • a2 • •••• ant··· (lim I ani = 00) n .... co

is equal to the order of the corresponding function nCr). PROOF. The series (1.16) can be rewritten as a Stieltjes integral in the following form: co

f

dn (I) ).

o

1

.

Integrating by parts we have3 (1.19)

If the series (1.16) converges, then both the positive terms on the right side of (1.19) are bounded, and since the second term is monotone increasing, the integral co

r ~dt 1),+1

(1.20)

•

o

converges. From this convergence it follows that for arbitrary r> ro(E)

or lim r .... co

nr) r

=

E

>0

and

o.

Thus the order of the function nCr) is not greater than the convergence exponent of the sequence. Also, from the above reasoning we see that conversely the • Here we use the fact that 11(1) == 0 for 0

< t < lall.

Sec. 4]

II

ESTIMATES FOR CANONICAL PRODUCTS

convergence of the integral (1.20) implies the convergence of the series (1.16).

Let now PI be the order of the function n(r). Then for asymptotically n(t)

E

>0

we have

< l'+2".•

Therefore for ), = PI + E the integral (1.20) converges, and therefore the series (1.16) also converges. Thus the convergence exponent is not greater than the order of the function n(r). The lemma is proved. To estimate the canonical product (U5) we first estimate the primary factors. LEMMA

2.

For p

> 0 and all complex numbers u

a

•

In I (a, p) I

where

< API1ul+ I+u I ' p

1

(1.21)

Ap= 3e(2+ Inp).

Eorp = 0

In Ia (a; 0) 1-< In (1 + Ia I). The second assertion of the lemma is obvious. Let p <; p/(p + I), then PROOF.

lui

(1.22)

> O.

If

uP } InIO(a; p)I=Re { In(l-a)+a+ 2u3 + .. , +p

~

u"

~ I u"I"

I u IP+l

,,+1

= -Re ~T-< ~-k- < (p+ 1)(1-lul) - p/(P + 1) the inequality In (1 + luI) <; lui leads to the following: lul lui" InIO(a; p)I-<2Ial+-2-+ ... +p 9

= Ia I" (~+ P 1 1 / ~ / + ... + ~ / ~ P

1'-2+ 2/ ~ r-

< I a r [2 (1+ ~ ),,-1 + ~! (1 + ~

1

)

r-"J

1

IP + 1

1u
3.

If the series (1.23)

12

GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

[CHAP. I

converges, then the infinite product co

IT (z)=

II a (~; p)

( 1.15)

an

71=1

satisfies in the entire complex plane the inequality r

InIII(Z)I
co

r ~(/~ dt+r r

n(t)\l ,1"+

Ir+

o

dt) (r=lzl).

(1.24)

I

r

in which PRooF.

p>O and k o = 1.

k,,=3e(p+l)(2+lnp)

for

From inequality (1.21) withp

> 0 we have

In I II (z)l

co

":1 I

r co

~

an

I"

rl1+1 ( I an I

+

A 11+1

r) -

pr

0

dn (t) r)

til (I

+

As )Vas proved above, it follows from the convergence of the series (1.23) that ·

nCr)

Ilin - - 1 r-+oo r P +

= 0.

Integrating by parts we have

f

co

InIII(z)I
rll+ 1 II

o

n(t)dt

pr+(p+l)1 IP+l (I r)2

+

r

co

< Ap(p + 1) r 11+1 .o

n (/)dt IP+1(/+r)

or r

Inin(z)l< Ap(p+l)r 1J

0')

(fo t:~~ dt+rJ t:~! dt). r

The case p = 0 is verified in a similar manner, replacing the estimate (1.21) by (l.22). THEOREM

4

(BoREL).

The order P of the canonical product co

IT (z) =

IT a ( :n ; p) 71=1

does not exceed the convergence exponent PI of the sequence {a .. }. PROOF.

converges.

The number p is the smallest integer for which the series (l.23)

Sec. 4]

13

ESTIMATES FOR CANONICAL PRODUCTS

Thus for the convergence exponent PI we have the inequality

P<'PI <,p+ 1. Suppose that PI 0 n(t)

< C·il..

From Lemma 3 we have r

In III (~)I

<Xl

< C>.kprp (f t>.-p-1dt+r f t>'-P-2 dt) = o

where B .. = kp

B).,C),r>.,

r

L,

1P

+ P + ~ _ A] ,

that is, the order 4 of II(z) does not exceed A, and therefore, it does not exceed Pl' If PI = P + 1, then, as was shown in the proof of Lemma 1, lim

t~co

~=O 1 tP +

and the integral <Xl

f o

n (t) dt

tP + 2

converges. Using this we have from the lemma that asymptotically In I II (z) I

that is, II(z) is at most of order P REMARK.

< erp + 1 ,

+ 1 and minimal type.

If PI is not an integer and if the upper density of the sequence

{an} is finite, then II(z) is at most of order PI and normal type; if PI is not an integer and if the density ~ of the sequence {an} is zero, then n(z) is at most of order PI and minimal type. This assertion follows easily from the asymptotic inequality n(t)

< (.l+e)tP.,

which, for PI not an integer, leads to the asymptotic inequality In I II (z) I

< (1 +e) Bp,rP ••

(1.25)

Note that we cannot obtain this inequality from Lemma 3 in case PI is an integer. Later on we shall see that for integral PI the assertion ceases to be true. For the further investigation of the connection between the growth of the function n(r) and the growth of In Mn(r) we require a series of general theorems that playa fundamental role in the theory of entire functions. • Clearly in this inequality one can replace 10(z)1 by Mn(r).

14

GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

[CHAP. I

5. Jenseo's Theorem THEOREM 5. Let fez) be holomorphic in a circle of radius R with center at the origin, and f(O) ~ O. Then R

2_

J

J IJ(Re'8) Id~

2~

n't(t) dt =

In

o

-In 11(0)1,

(1.26)

0

Izl < t. the.circle Izl < R,

where n,(t) is the number of zeros offez) in the circle

Note that if the function has no zeros in then net) = 0 and equation (1.26) expresses a well-known property of harmonic functions. If there are zeros, then it follows from (1.26) that

..

2~

In 1/(0) 1<

fin II(Re &) I d6. i

D

PROOF. If the function has no zeros on the circle argument principle we have

n, - f

Izl =

t, then by the

20<

1 (t) _ _ 2ft

d[arg/(te'8)] dO

6 d .

o

Using the Cauchy-Riemann equations 0[arg/(te'8)] _

toO

-

we have

olnl/(teil)1 ot

2K

= 2~

J

:1 In IJ(te il ) I d6 o for all except a finite set of values of t in the interval 0 < t < R. Integrating both sides of this equation from 0 to R we obtain (1.26). Here, of course, we are using the continuity of the function n't(t)

2K

fin II (te i ') I d6. o It is easy to establish this continuity if we represent the function In If(z)1 . in the form In Ih(z)1 + l:1..I<;R In Iz - atl, where at are the zeros of the function fez), andfl(z) has no zeros in the circle Izl <; R. The first term is continuous, and

fo In I Re III

iB

- ak ld6

=

{

2'11: In R 2'11: In lakl

for

R> lakl,

for R

< 10k!'

so that this is also a continuous function. Thus the theorem is proved.

Sec. 5]

15

JENSEN'S THEOREM

Jensen's formula can be generalized to include the case when J(z) has a zero of order A. at the origin. Then, applying formula (1.26) to the function J(z)z-A, we obtain .R

AIn R

+ Jn,t) dt = o

f

21<

2~

In If(Re i9 )1 de -In

1t<~I(O) I .

0

Here nf(t) is defined to be the number of zeros of J(z}, different from the origin, in the circle Izl < t. The following lemma gives an important estimate for the number of zeros of J(z} in a circle. LEMMA

4.

IJJ(z} is holomorphic in the circle

Izi <,.er and ifIJ(O)II,

=

then

(1.27) PROOF.

From Jensen's formula and the monotonicity of nAt) we have at

once, 21<

tW

n,(r}<.

Sn,/,) dt<. 2~ flnlf(erei6}ldO
(1.28)

0

Let us investigate when the equality sign can hold in (1.27). For this it is necessary and sufficient that equality hold everywhere in (1.28). The equation

f f!r

n,(t)

n,(r) = . - t - dt r

says that the functionJ(z) has no zeros in the annulus r The equation er

f

< Izl < er.

21<

n,(t) -'-t-

r

1 dt = 27t

SIn If (ere'&). IdO 0

=

says, by (1.26) since IJ(O)I Finally, the equation

1, thatJ(z) has no zeros in the circle Izl

k

2~

J If(erei~} Id~ = In

In M,(er)

o

says that on the whole circumference Izl = er

I f(z)1 = M,(er).

< r.

16

GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

[CHAP. I

All these conditions are fulfilled for functions of the form 5

( lakl =

r,

is real ).

Gt

Indeed, each factor in this product maps the circle 1=1 ~ er onto the unit circle, and therefore has modulus one everywhere on the circumference 1=1 = er. Thus on the circumference 1:1 = er we have 1
PROOF. Without loss of generality we may assume that f(O) wise in place of f(z) one considers

f

(z) 1

=

A!Z-A

=

1. Other-

f(z) ftA) (0) ,

which has the same order and convergence exponent as f(z). From (1.27) it follows that

n,

-.- In (r) -.Pl= hm Inr
ro+oo

In In Mr (er) 1 =p.

!l er

The theorem is proved. Note that from (1.27) one can also obtain -

n (r)

A, = lim - r 0+ co r P' and if PI

In M (er)
= P we will have ll,

<e

POf'

Comparing this with Theorem 4 and the remarks following it we have the following theorem. THEOREM 7. For canonical products the convergence exponent of the zeros is equal to the order of the function. In addition, if the convergence exponent is not an integer, then the canonical product is of maximal, minimal or normal type according to whether the upper density of the set of zeros

A,=

n,(r) lim - ro+co

rP

is equal to infinity, to zero, or to a number different from zero and infinity. • It is not difficult to show that this is the general form of the functions for which equality holds in (1.27).

Sec. 6]

17

THE MAXIMUM MODULUS OF A HOLOMORPHIC FUNCTION

Later we shall investigate more fully the connection between the type of an entire function and the quantity /).,.

6. The Relation between the Maximum Modulus of a Holomorphic Function and the Maximum of Its Real Part We denote the maximum of the real part of a functionf(z), holomorphic in the circle Izl ~ r, by A,(r). From the maximum principle for harmonic functions it follows that A,(r) is a monotone increasing function of r. Also we clearly have I A,(r) 1<- M,(r). For R > r there is an inequality between A,(R) and M,(r), which is a sort of converse to the inequality above. THEOREM 8.

If f(z) is any function holomorphic in the circles Izl

M,(r) <- (A,(R)- Ref(O») R 2r _ r

+ If(O) I

(r

< R).

~

R, then (1.29)

This inequality is called the Caratheodory inequality for the circle. PROOF.

Our starting point is the formula of Schwarz,

r"

i<\!

1 f(z) = -2 u(Re i <\!) Re.<\! + z d'?+ Lv (0) 1t . Ret -z

( 1.30)

-"

(f(z) = u (z)+ Lv (z». which represents a function holomorphic in the the circle Izl <; R in terms of the boundary values of its real part. 7 Let us add to the right side of (1.30) the quantity

u (0) -

2~

.

f

U

(Re ttJI ) d'?,

which is equal to zero. We obtain the equation

.

1 f(z)=-1t

f

u(Re'tJI)

z

Rei<\!-z

d'?+f(~.

In particular, for f(z) == 1 we obtain

.

1 1t

f

z

Re'<\!-z

d"

=

O.

-l<

• This hypothesis can be weakened slightly. It is sufficient to require that f(z) be holomorphic in the circle Izl < R and that its real part have a limiting value at each boundary point, and that these boundary values be bounded above. 1 The derivation of formula (1.30) may be found, for example, in Titchmarsh (I).

18

GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

[CHAP. I

From the last two equations we obtain the formula

"

J' (A,~R)-u(Ref"'») Re''''-z dl~-f(O).

-f(z)=1 K

"z

-"

Since the quantity A,(R) - u(Re'''') is non-negative we obtain the inequality

I/(z) I

< R2r_ r [A,(R) -

u (0»)

+ I f(O) I·

Clearly it is possible to replace If(z)1 by M,(r) and obtain (1.29). If the function w = fez) takes values lying in the left half-plane (Re w < 0) when z is in the circle Izl < R, then AAR) < 0 and

I/(Z)I<-R 2r r u(O)+I/(O)1 (lzl 0 satisfies, for 1m z and Izl > 1, the inequalities

~ I/(i)1 51

;0 0 >0

(1.31)

< 'It).

This inequality will be called the CaratModory inequality for the half-plane. PROOF.

Map the upper half-plane 1m z

z-i u = -z+i

or

Z

> 0 onto the unit circle

=-

a+l a-I

1--.

The function

F (u) = 11(- i a-I a + 1) is defined in the unit circle and satisfies Re F(u) <; O. Consequently, A P{R) <; 0, and from Caratheodory's inequality for the circle lui <; I we have

I fez) I -<: I I(i) I - Re (il(i» I z +~II ~I:~ i I Note that for

Izl :>

(1m z

> 0).

1

Iz -II -<: 2r. I z + i I ~ 2r

and I z

+i1

9 -/

z - i 12 = 4r sin Ij, •

• It is easy to obtain the last equality, if we consider the triangles with vertices "at 0, z, i and at 0, z, -I.

Sec. 7]

19

THE MODULUS OF A POLYNOMIAL

we obtain

1/ (z) I < 1/ (i) I (I

+ 5~~ 0 ) < 51! (i) 1 51: U •

(1.31 ')

To obtain the left side of the inequality note that the functionf(z) does not vanish in the open upper half-plane. Writing the inequality (1.31') for the functionf- 1(z) we obtain the left inequality in (1.31). Theorem 8 enables us to estimate from below the modulus of the holomorphic function f(z) having no zeros in the circle Izl <;; R. Indeed, assume thatf(O) = I, and write inequality (1.29) for the holomorphic function Inf(z). We obtain the inequality

Ilnl(z)I
(O
from which follows

I

In /;Z) or

1-< R 2r r In M,(R).

2r In 1/(z)I>- R_rlnM,(R).

Thus we obtain the following theorem. THEOREM 9. If the function f(z) is holomorphic in the circle Izl <;; Rand has no zeros in this circle, and if f(O) = I, then its modulus in the circle Izl <;; r < R satisfies the inequality

In II(z) I

>-

2r R_r ln M,(R)

(I zl <: r

< R).

(1.32)

If the function f (z) has zeros, such an estimate is, of course, impossible. However one can give an analogous estimate in the domain obtained from the original circle by removing certain small circles containing the zeros. To obtain this precise estimate, which plays an important role in the theory of entire functions, we first estimate from below the modulus of a polynomial.

7. Lower Bounds for the Modulus of • Polynomial The following theorem of H. Cartan [1] provides an estimate from below for the modulus of a polynomial. THEOREM 10. Given any number H > 0 and complex numbers a 1• a2 , ••• , a.., there is a system of circles in the complex plane, with the sum of the radii equal to lB, such that for each point z lying outside these circles one has the inequality

Iz-a1 1·lz-a2 1·· ·Iz-alll > (~r.

(1.33)

20

GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

[CHAP. I

We divide the proof into several stages. I. We choose the quantity H/n as the unit of measurement and show that there are closed circles in the complex plane, having radius equal to the number of points {at} contained within the circle. Indeed, form the smallest convex polygon containing all the points {a k }, and choose any vertex aj of this polygon. Clearly there are circles of arbitrary radius that contain this point but do not contain any other of the points {at}. In particular, the radius can be chosen to equal the multiplicity of the point a j • 2. From among all circles with radius equal to the number of points {at} lying inside the circle choose one with the largest radius, )..lH/n, and call it C1• Note that no circle in the plane with radius greater than or equal to )..lH/n can contain more points of the set {at} than the number of units of measurement in its radius. Indeed, suppose a circle of radius )"H/n, with), ~ )..1' contains)'" > ).. points of the set {a k }. The concentric circle of radil!s )..' H/n either contains A' points or ).." > )..' points. In the second case we consider the concentric circle of radius A"H/n and so forth. Since the set {a k } is finit P , we eventually come to a circle of radius )"H/n, larger than A111/n, that contains A points. This is impossible, since C1 is the largest circle having this property. The points of the set {at} that lie inside C1 will be said to be of rank AI. 3. We remove the points of rank )..1' and for the remaining n - ;'1 points we construct the largest circle Cz that contains the same number of points as there are units in its radius. Let its radius be )"zH/n. We shall show that )..2 < AI. Indeed, if this were not the case then the circle Cz would have a radius larger than A1H/n, and would contain at least as many of the original n points as there are units in its radius. But this contradicts the result of §2. The points in Cz will be said to be of rank Az. Now remove these points too, and for the remaining n - Al - Az points we find the largest circle C3 containing the same number of points as there are units in its radius. Suppose that its radius is laH/n. Clearly Aa "All· The points in C3 will be said to be of rank )..3' and so forth. We thus obtain a sequence of circles PROOF.

C1 , Ci ,

.•. ,

Cp

with radii that contain AI, All, ... ,A" units of measurement, respectively, where Al ;;> ~ ;> ... ;;> ;." and H ,,(A 1 +A i + ... +J..p)=H.

4. We now form circles

r l • ri ,

.•••

rp

concentric with the circles C1 , Cz, ... , C" but with radii twice as large. Let z be an arbitrary point lying outside all of the new circles. Describe about z the circle C. of radius )"H/n, where A is some natural number. This circle does not

Sec. 8]

21

THE MODULUS OF A HOLOMORPHIC FUNCTION

intersect any of the circles Cj that have a radius larger than or equal to A. Thus this circle can only contain points whose rank is less than A. From the definition of rank it follows that after removing all points of rank greater than or equal to A, no circle of radius greater than or equal to AH/n can contain as many of the remaining points as there are units of measurement in its radius. It follows from §2 that the circle C. can contain at most A-I points. 5. Enumerating the points {a k } in the order of increasing distance from z we have and

8. Lower Bounds for the Modulus of a Holomorpbic Function THEOREM 11. Let fez) be h%morphic in the circle Izl ~ 2eR (R > 0) with f(O) = 1, and let 1'} be an arbitrary positive number not exceeding 3e/2. Then inside the circle Izl ~ R, but outside of a family of excluded circles the sum of whose radii is not greater than 41'} R, we hat'e

In

for

If(z)1 > -H(Tj) In M (2eR) H (Tj) = 2

PROOF.

Q2"

+ In 3e2-tj'

Construct the function

cp (z) = where a l ,

(1.34)

(-2R)" altZ:i .•• a"

" 2R (z-ale) II , Ie

=1

(2R)2 -

an are the roots ofj.(z) in the circle

•• ,

cp (0) = 1 and Icp (2Re i8 ) I =

De function IJI(Z)

K

alez

Izl < 2R. We have

(2R)"

I altZ:i ••• a" I

•

= f(z)

,00

_ no roots in the circle Izl ~ 2R, and therefore by the corollary to the Caratheodory theorem for the circle (Theorem 9) we have for Izl ~ R In

IHz) I> -

>-

21n M,(2eR)+ 21n i al~2~)."a" I 21n M,(2eR).

Now we estimate I!p(z) I from below. For Izl :( R

II" I (2R)9 -

a"z I < (6R'l)fI •

(1.35)

22

<.iENERAl THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

[CHAP. I

Applying the Cartan estimate (see Theorem 10) to the polynomial in the numerator of 'P(z) we see that, outside of a family of circles with the sum of the radii equal to 47JR, we have

" III 2R (z k

ak)

~1

I>

e:

R )" (2R)"

and consequently

(2R)2"

(2"1R)"

I~ (z) I> I al~ ... a" I e

1

(2'1 )"

(6R2)" ~ 3e

•

By Lemma 4 n = n (2R) " In M,(2eR). ~nd

for

1'P(z) 1this leads to the estimate In I!,? (z) I

> In (~) In M,(2eR).

Combining this with the estimate (1.35) we see that in the circle 1=1 <; R, but outside of a family of circles, the sum of whose radii does not exceed 41JR, we have In If(z) I (2+ln ~~)In M(2eR).

>-

9. The Growth of tbe Product of Two Entire Functions It follows at once from the definition of the order of a function that the order of a product does not exceed the larger of the orders of the factors. With the aid of the lower bound for the modulus of an entire function that was established in the previous section, we can obtain a much more precise result. We have the following theorem. THEOREM 12. (a) The order of the product of two entire functions of difJere1/t orders is equal to the larger of the orders of the factors, and the type is equal to the type of the function that has the larger order. (b) The product of two entire functions of the same order, one of which has normal type (1 and the other has minimal type, is an entire function of the same order and of type (1. (c) The product of two entire functions of the same order p, one hat'ing maximal type and the other hat'ing at most normal type, is an entire function of order p and of maximal type.

This theorem can be formulated more briefly by introducing the notion of category. We shall say that the functions fez) and 'P(z) belong to the same category if they have the same order, and if they are both of maximal type, or both of normal type, or both of minimal type. Otherwise we shall say that the function having the larger order, or if the orders are equal the one having the larger type, is in the larger category.

Sec. 9]

THE GROWTH OF THE PRODUCT OF TWO ENTIRE FUNCTIONS

23

Theorem 12 may now be formulated as follows:

If the function fez) and tp(z) are of different categories, then the product fez) tp(z) has the same order and type as the factor belonging to the larger category. PROOF. We limit ourselves to the proof of (b), since the other two parts are proven similarly. Letf(z) be an entire function of order p and type a > 0, and let tp(z) have order p and minimal type. From the asymptotic inequalities

M,(r)

< e(·+i),.P

and M,(r)

< e: ,.P

we have the asymptotic inequality M~ (r)

< etH1) rP •

(1.36)

To obtain the necessary estimates from below for the function M,,,,(r) on some sequence of values of r tending to infinity, we apply to tp(z) the lower bound found in § 8 (Theorem II). Here we may assume without loss of generality that CP(O) = I, since multiplication of tp(z) by cz- n changes neither the order nor the type of the function. First we find a number Rl (arbitrarily large) such that

> e{Ir-j-) B~

M,(R1>

and for all R ;> Rl M, (R)

< e&B

p•

Such a choice is possible for arbitrarily given positive E and d since the first inequality is satisfied on some sequence tending to infinity, and the second holds asymptotically. We shall assume that 0 < d < I. Choose 7J = d/8 and inside the circle Izl <; R (R = R1(l - d)-I) form the excluded circles described in Theorem II. The sum of the diameters of all the excluded circles is less than dR. Therefore in the interval (R 1, R) there is a number r1 such that the circumference Izl = r1 .does not meet any of the excluded circles. From Theorem lion this circumference we have

In IIf (z) I > - (2+ In 1~) In M, (2eR). Since Rl

< r1 < R1(l M,(r1)

- d)-I, we have " ) taP P > M,(R1) > e( "-2"• ) B1P > e("-2" \ - ) "1.

Comparing this with (1.37) we have

In M,.(r1)> [(0- ; )(l-Wr~ -(2+ In 1~) In M,(2eR)].

(1.37)

24

GENERAL THEORY OF THE GROWTH OF ENTIRE fUNCTIONS

[CHAP. I

and since we have

In M,,(r1)> [(0- ;)(1-3>'-(2+ In 1~)(1-o)-P(2e>,o]r~. For given E > 0 one can choose 6 so small that the expression in the square brackets is not less than (1 - E. Consequently,

M,,(r 1)

> e("-·l~

for a sequence of values r 1 tending to infinity, and the theorem is proved. COROLLARY. If the quotient of two entire functions f(z) and !p(z) is an entire function lp(z), then its category does not exceed the larger of the categories of the functions fez) and !p{z); here the categories of fez) and !p{z) may be the same. If they have different categories, then the category of the function lp(z) equals the larger of the categories offez) and !p{z). PROOF.

We are given

cp (z) ~ (z) = ! (z). If the category of lp(z) is larger than the category of !p{z), then the category off(z) is equal to the category of lJ'{z). Thus the category of lp(z) cannot exceed the categories of fez) and !p(z), and the first part of the corollary is proved. Clearly also, if the category off(z) is larger than the category of !p{z), then the categories of fez) and lp(z) are equal. On the other hand, the category of fez) can be smaller than the category of !p(z) only in case the categories of !p{z) and 1p{z) are equal. 10. Hadamard's Theorem

The theorems of the preceding sections enable us to refine considerably the theorem on the representation of an entire function as an infinite product. This refinement, which is due to Hadamard, concerns the representation of entire functions of finite order, and is one of the classical theorems of the theory of entire functions. THEOREM

13. The entire function f (z) offinite order p can be represented in

the form

•

!(z)=z"'eP(.1

II a (.!....; p) lin

«(1)-<00),

(1.38)

n=1

where an are the nonzero roots offez), p <; p, P(z) is a polynomial whose degree q does not exceed p, and m is the multiplicity of the zero at the origin.

This theorem can also be formulated as follows: The genus of an entire function does not exceed its order (for the definition of genus, see the next to the last paragraph in § 3).

Sec. 10]

2S

HADAMARD'S THEOREM

PROOF. The convergence exponent PI of the zeros of the entire function does not exceed its order P, and the integer p in the associated canonical product

...

IT (z) =

II a (:n ; p)

«(I)

-< 00)

11::1

does not exceed Pl. Consequently, p <; p. It remains to show that the function g(z) in (1.14) is a polynomial of degree q not exceeding p. To this end we recall that the order of the canonical product is equal to the convergence exponent PI <; p. By the corollary to Theorem 12 the entire function. Hz) = f(z) n- 1 (z) Z-III has order at most P, and therefore satisfies the asymptotic inequality

+·.

In I ~(z) 1< r P

In addition, the function 1p(z) has no zeros. Thus g(z) = In ~(z)

is an entire function, and we have asymptotically Ag(r)

< rPH.

By the Caratheodory inequality it follows that asymptotically Mg(r)

< r pH•.

By Theorem I the function g(z) is a polynomial of degree at most p. The theorem is proved. For functions of nonintegral order the degree of the polynomial in the exponential factor is less than p. Therefore the order of the function coincides with the order of the canonical product, and therefore with the convergence exponent. Thus in this case ',,".,

and the genus of the function is p = [p]. For integral P the genus of the function is either equal to P or to P - I. Indeed, if the genus is less than p, then q < P and the order of the function coincides with the order of the canonical product, which is equal to the con'Pfgence exponent Pl . . ; On the other hand, p <; PI <; P + I, so that p <; P <; P + I, and since by Msumption p < p, we have p = P - 1. EXAMPLES.

The function fez)

= sIn Yz 'ltYz 'It

26

GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

[CHAP. I

is entire, of order one-half, and has the zeros ra, 21, • • •• By Hadamard's theorem it is of genus zero and therefore has a representation

c!J ( 00

f(z)

=

I -

:~).

From the equationf(O) = I we have c = l. Replacing z by Zl we obtain co

sin'ftZ='ltZ

IT (I - :a), ,,=1

from which we also have co

•

sin'ftz='ftzII'(I- :)en . -co

Thus sin 7TZ is not only an entire function of order one, but also of genus one. Similarly, for the entire function of order one _ 1 ( ....... f(z)-n e +e

+... + e..,,-) ,

where Wi (j = 1,2, ... ,n) are all the roots of the equation w" obtain the expansion

+ 1 = 0,

we

co

f(z)

= 11(1- ~). It-I

Clot

in which exl: are the zeros off (z). It is easy to show that this function also has genus one. The function

is entire of genus zero. On the other hand the convergence exponent is one, and therefore it is of order one. In general, if the convergence exponent PI of the sequence {a .. } is an integer and if the series

diverges, then the genus p .of the canonical product is equal to its order P = Pl. If the series converges, then the genus of the canonical product is equal to P - I. We have already observed that functions of nonintegral order have an infinite set of zeros, and the convergence exponent of the set of zeros is equal to the order of the function. In this case, by assertion (a) of Theorem 12, the function/(z) has the same order and type as the canonical product. From this remark and Theorem 7 we have the following more precise theorem.

Sec. 11]

ENTIRE FUNCTIONS OF INTEGRAL ORDER

27

14. If the order p of the entire function f(z) is not an integer, and then for tl, = 0 the function is of minimal type,for tl, :;. 0, 00 the function is of normal type, while for tl, = 00 'it is of maximal type. THEOREM

if tl, is the upper density of the set of its zeros,

Since the order and type of the function f(z) - a coincide with the order and type off(z) for all complex a, it follows from Theorem 14 that a function of nonintegral order takes on all complex values "equally often." 11. Entire Functions of Integral Order The conclusion of Theorem 14 is no longer valid when the order of the entire function is equal to an integer. Indeed, in this case the order can be determined by the exponential factor, and the convergence exponent may be less than the order of the function. In particular, the function need not have any zeros at all. But this is not the only peculiarity of functions of integral order. It turns out that in this case the type of the canonical product is not determined just by the upper density tl of the zeros, but depends also on the distribution of the arguments of the zeros. Two examples that illustrate this are the functions sin ;

z=; zIi (1-(2!;) n~1

and IX)

~ (z) = r ~Z) = ze yz II (1 n-l

+ ~) e - ~ •

for both functions we have n(r)"'""-'r,

that is, the density of the zeros is the same. However, sin (7Tz/2) is of order one and normal type, while 'P(z) is of order one and maximal type. This is easily ICen for the latter function by using Stirling's formula 9

-Inr(z)=(z- ~)Inz-z+ ~ In21t+O(}), : from which it follows that

In M.,(r)"'""-'r In r. • This formula is usually established for all complex values of z lying in some angle "1T - 6 (see, for example, Titchmarsh[1]). However, from the relation r(z)r(l - z) = !/Sin 1TZ one easily sees that it is valid outside of circles of arbitrarily small constant radius, ibout the points 0, -I, -2, .... This is sufficient for our conclusion.

largzl

28

GENERAL THEORY OF THI: GROWTH OF ENTIRE fUNCTIONS

[CHAP. I

In the investigation of functions of integral order p the quantity

o,(r)

=

+

IXp

+~

a;P

I an I "',. plays an essential role, where p is the order of f(z), and oc p is the coefficient of zP in the exponential factor in the representation (1.38). Let

The question of the type of an entire function of integral order is settled by the following theorem of Lindelof [1]. THEOREM 15. (a) If the genus p of the canonical product of the entire function f(z) of integral order represented by (1.38) is equal to its order p I, then for Yt = 0 the function f(z) is of minimal type,for 0 < Yt < 00 of normal type, and for Yt = 00 of maximal type. (b) If the genus of the entire function is less than its urder, then its type is equal to the coefficient of zP in the exponential factor standing in front of the canonical product.

>

PROOF.

=

Consider first the case p fR (z) =

p. We introduce the function

IT a (a:; P-

1)

II a (a:; p),

( 1.39)

I an I> R

I an I" R

with the aid ofwhichf(z) can be represented in the form

() f( z-ze ) - m Pp-l(') e"f(R),P f RZ'

( 1.40)

where P,,-1(:::) is a polynomial of degree at most p - I. T~is important representation will be used frequently in the study of entire functions of integral order. To estimate the growth of the function(u(:::) we use inequalities (1.21) and (1.22). Assume that p > I. Then, if we put MR (r) =

max

IfR (rei~) I.

0.,;;:8<2"

we have

R In MR (R)

[Rpf p

o

+ Rp+ t .rtP(t+R) dn (t) 1. tP-1(t+R) en

dn (I)

R

Just as in the proof of Lemma 3, if we integrate by parts we will obtain the inequality R 00 In MR (R)

J

< kp [ RP- 1 f n t~) dt+ R P+ 1 ;p~~ dt]. o

in which kp is a constant. In case p similar manner.

=

(1.41)

R

I, inequality (1.41) can be obtained in.

Sec. II]

29

ENTIRE FUNCTIONS OF INTEGRAL ORDER

If I:!., is the upper density of the set of zeros of fez). then asymptotically for

e>O net)

< (6,+!)t~.

and from (1.41) we have the asymptotic inequality In MB (R) Also, from (1.40) we have

< 2k p(6,+a) RP•

In M,(R) < I3,(R) IRP

+ In MB (R)+ 0 (R

( 1.42) P- 1).

(1.43)

and, taking account of (1.42), we obtain the asymptotic inequalitylO

In~(R) < I0r(R)I+ 2kp (6 r +a)+O(!) or

Or -<. (1 + 2k,) T,.

(1.44)

To prove the reverse inequality we note that by Lemm.\ 4 lim n, (r) rP

-<.

,. ... 00

lim e P In Mf(er) (er)P

,. ... 00

and, consequently,

(1.45) It remains to give an analogous estimate for (J,. For this purpose we must estimate the functionfR(z) from below. This can be done using Theorem 11. Choose 1J = i e-3 • Then inside the circle Izi r but outside a set of excluded circles, the sum of whose radii is less than 6re- 3• we have the inequality

<

In IIB(z) 1> -H(i ,-S)In MB (R)=-6InMB(R). where R = 2er. Since 6e-3 < i. there is a number r1 between r/2 and r such that the circumference Izi = r1 does not meet any of the excluded circles. From the represen·iation (1.40) we see that on this whole circumference we have the inequality

Re (or(R)zP)

< 61n MB(R)+ln M,(r1) +O(R,-l).

, Choose the argument of z so that zP and t5,(R) have arguments that cancel 9ne another. Then we will have

(1.46)

1. The direct application of (1.24) to the estimation of a canonical product of integral order

, leads to the estimate

In M, (R)

< D, (Il, + .) Rf In R.

30

GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

Since'l

[CHAP. I

> (4e)-IR, it follows from (1.42) that InMR (R) lim R-+oo r~

< 2k/J., (4e) P.

( 1.47)

It follows from (1.45), (1.46) and (1.47) that

'0,< Co"

(1.48)

where

C= 12kp(4e'.!)P+ 1. Combining (1.45) and (1.48) we obtain

(1.49)

0t:;:? C-l 1,·

The assertion of the theorem in case (a) now follows from inequalities (1.44) and (1.4Q). Now suppose that p < p. In this case the series

~ la:"p converges. Therefore the integral 00

r

n (t) • t P+ 1

dt

(1.50)

o

also converges and, in addition, lim n (I) = O. '-+00

(1.51)

tP

Applying Lemma 3 to the canonical product TI(z) of the functionf(z) we obtain the estimate r

00

InITI(Z)I
< a,',

which means that the function TI(z) grows no faster than a function of order p and minimal type. By Theorem 12 (b), it follows from this that the type of the functionf(z) coincides with the type of the exponential factor. that is, it is equal to the coefficient of zP in this factor. Thus if the genus of an entire function is less than its order, then (when the order is an integer) it must be of minimal type. In conclusion, we note the meaning of the conditions lim r-+oo

I 'O,(r) I < 00,

lim 8,(r) = 0, r-+oo

Sec. 12]

31

PROXIMATE ORDERS

that play an essential role for the case p == p. In the presence ofthese equations the series

~la:IP diverges. Hence 6,(r) remains bounded only if there is some special symmetry in the distribution of the roots a". In the absence of such symmetry lim sup 16,(r)1 == 00. For example, if all the roots a" lie on one ray, then 6/(r) -+ 00. In this case fez) is of maximal type, by what has been proven. This was the 1(z). Thus, for case with the example we considered before of the function functions of nonintegral order, the type is determined by the modulus of the zeros, while, for functions of integral order, the type also depends on the di~tribution of the arguments of the zeros.

r-

12. Proximate Orders The growth scale that we have used in the preceding sections can be further refined. Thus, for example, Lindelof compared In M,(r) with functions of the form II , PI nII, , 1n.lit , . .. Inpl", where In;r == In Ini-lr and atl' Ott, ••• , IX" are real numbers. Clearly each successive logarithm grows more slowly than any power. of the preceding one. The generalized order of the function fez) is defined to be a system of numbers p, atl' Ott, ••• , at" such that for arbitrary E > 0

,P loll, , Ini', ... In;1I

-I, < In M, (r) < r

P lo"'

, lo~' , ... 10;11+0 r

(a> 0).

in which the right inequality is asymptotic. while the left inequality is satisfied (or some sequence of values of r tending to infinity. The order p', at~. ~, ••• , at;, of the function Ji(z) is said to be greater than the order p, lXI' lXa, ••• , at" of the ·.. functionf(z), if p' > p, or if p' == p and at~ > atl' or more generally if p' == p, • ~ == atl' ••• , 1X1 -1 == at;_l but IX} > atl' All the theorems of the preceding llections can be generalized to this more precise growth scale. In particular, it can be shown that if p is not an integer, then the asymptotic inequalities

nr(') ~

< ,P In'" T

•••

In;II+o'

equivalent, and therefore in this case the generalized orders of the functions lItCr) and n,(r) are equal. . We shall not prove these theorems. as we shall soon prove more general . theorems. The growth scale can be further refined by introducing as comparison

32

GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

[CHAP. I

functions monotone functions that grow more slowly than any of the iterated logarithms In,r, etc. However, instead of such further refinements, it is more natural to define in a general manner a certain class of "slowly increasing" functions L(r) and then to compare In M,(r) with functions of the form rPL(r). This was the path taken by Valiron [1], who introduced the concept of proximate order of growth. A function p(r) that satisfies the conditions l l lim p(r)

= p ;;> 0

and lim rp' (r) In r

=0

(1.52)

is called a proximate order. If for the entire function f(z) the quantity - . InM,(r) 0,= , lim ( ... co r P r) is different from zero and infinity, then p(r) is called a proximate order of the given entire function f(z), and at is called the type of the function f(z) with respect to the proximate order p(r). Clearly, the proximate order and the corresponding type of the given function are not uniquely determined. For example, if we add In clln r to the proximate order, then we obtain a new proximate order for the given function, and now the type has been divided by c. With respect to an arbitrary proximate order, a function may be of minimal, normal, or maximal type. With respect to its own proximate order it is always of normal type. A positive function will be called slowly increasing, and will be denoted by L(r), if lim L (kr) = 1 , ... co

L (r)

uniformly on each interval 0 < a ~ k 0( b < 00. The following lemma plays a fundamental role in the study of proximate orders. LEMMA

5.

If p(r) is a proximate order, then the function r P(,)-p

is slowly increasing. 12 PROOF.

then

Let L (kr)

In L(r) =(p(kr)-p)lnk+(p(kr)-p(r»)lnr. II Here and throughout this book the assertion lim cp(r) = A means "the limit exists and is equal to A." 11 The converse is true in the presence of certain supplementary restrictions on the function 1.(r) (for example. if 1.(r) is logarithmically convex).

Sec. 12]

33

PROXIMATE ORDERS

To fix ideas let us assume that 0 < (/ .;;; k < I. Then by the second of conditions (1.52), for arbitrary 1] > 0 and all sufficiently large values of r we have by Lagrange's theorem:

Ip(r) -

p (kr) I <

~ ~ -<: '1l

1j ;,.

(+ - 1)

In

:ar) .

Also, for r sufficiently large we have by the first of conditions (1.52)

I p(kr)-p 1< 1j and thus uniformly for 0

< (/ .;;; k

.;;; 1

· InL -(kr) I1m -= 0

f'-+OO

L (r)

L (kr)

or

lim -L() =1. ,.

f'-+OO

The case k > I is treated similarly. We now establish some simple properties of proximate orders that we shall need frequently. (I) For p > 0 the function r"'-r) is monotone increasing for all suffiCiently large values of r. Indeed, (rWI)' = p (r) r P(f',-1+ rP("!p' (r) In r. and since the quantity rp'(r) In r tends to zero we have asymptotically (rP(f'»'

> (p-a)rP(f')-1 > 0

(0
Note that the assertion remains true for p = 0, provided that p(r) > 0 and the second condition of (1.52) is replaced by the condition p'(r)rln r/ p(r) - 0 as r - 00 (for p > 0 these conditions are equivalent). Indeed, in this case p(r) In r - 00 and (rP(f'»'

= p(r) rP(f')-1 (I +

~(~) r In

r) > p(r)

rP(f')-l (1

- It)

> O.

Since in the study of asymptotic properties the behavior of the proximate order on a finite interval plays no role, we shall in the future assume that rp(r) is a monotone function for r ;> 0, and we shall define it to be zero at the origin. This may be done without loss of generality. (2) For r - 00 and 0 < a .;;; k .;;; b < 00 the asymptotic inequality (I-e)k'r'(f')

< (krY

(kr)

< (I +e)k'rP(r)

Mitis uniformly in k. This inequality follows at once from Lemma 5 and the definition of slowly increasing function. Note that for k > 1 we have the asymptotic inequality (krY (1:,,)

< kPHr' (r) •

34

GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

+1

(3) For A < p

,..

tPltl-), dt =

f

[CHAP. I

1 P(rJ+l-),+ or (p(rl+l-).) . p +1 - A·r

( 1.53)

II

Indeed, integrating by parts we have

r. r

1 [tPUl-), (p (r) - p) p+l-)."

-

+ t P(tl+l-V (t) In t] dt.

From the definition of proximate order we have asymptotically

Ip(r) - pi < -;. ,

lip' (I) In t I <

;.

Using this we have r

(I +O(I»ftP(tl-l. dt =

1 rP(r)+l-).+O(I) p+l-A

II

and, finally,

f

r

tP(t l -). dt = p

+ ~ _ A rP(r)+l-). + °(rP(rl+l-).).

II

For A > p

+ lone obtains in a similar manner the formula co

f

1 rp(rl-A+l+o(rp(rl-).-l). A-p-l

tP(tl-).dt=.

r

(1.53')

(4) I/fP(r) is a bounded/unction on each finite interval, then/or A 0).

where A __

~

Indeed. for

E

> 0 and r > r. 'P(t)

-I.

,,(r)

r .. co

rf (r)

1m - - .

< (A+a)tP(t)

and consequently r

r

S";t) dt ~O(l)+(K+a) Sfl'l-).dt. •

r.

+1

Sec. 12]

35

PROXIMATE ORDERS

Applying Rule (3), we obtain

f ,( 1"

t)

-

t).

-

dt ~ ~

(11

rP + I) p+l-A. + 0 (rP (1")+1-).

(1")+1-).).

II

From this we have at once

lim 1"-ho

{r-

r

p (1")-1+A

f ,~) t

dt}

-< p+A.-l ~ .

II

Similarly one sees that

lim

-

r

{r- p

(1")-1+).S

'(t)dt}........

A -?'p+l-A.'

f

•

r~oo

where A = Urn ,(r) •

-

-

r+oo

rPl1" )

From Rule (4) and the last remark we have at once the following assertion: (5) If cp{t) is a bounded function on each finite interval and if the limit

11 = lim ,(r) r+oo rP (1")

exists, then for .A.

the limit

f ,~t) r

lim {r-p (r)-1+)..

1"~oo

t

II

dt}

=

A p+l-A.

exists. Valiron established the basic fact that every entire function has a proximate order. Thus after we have introduced the concept of proximate order, every function of finite order can be regarded as a function of normal type. We prove a somewhat more general theorem. THEOREM 16. If f(r) is a~ continuous junction that is positive for r 8IlIisfies the condition

-

p= lim

r~oo

> 0 and

ln/(r) lnr

--<00,

then the proximate order p(r) can be chosen Sf) that for all positive values ofr /(r)-< r Plr)

ttnd fOT some sequence of values r.. (n

= 1, 2, ... ) tending to infinity

f(r,J = rt(1",,). 1. In place ofthe given functionf{r) it will be more convenient for .. to consider the function PROOF.

,(r)=/(r)rp,

36

GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

[CHAP. I

whose growth is less than any power of r. Clearly -;- In, (r) 1 1m - - = r+oo Inr

0

.

Passing to the logarithmic scale, that is, putting x = In rand y = In 9"(r), we have 'PI (x). where Ci'l(X) = In Ci'(ez). At the same time the equation

.v=

lim 't(x) 11:+00

=0

X

is valid, from which it follows that for arbitrary e > 0 and for all values of x that exceed a certain x., the entire curve y = Ci'l{X) lies below the line

y=ex. On the other hand there are points on this curve with arbitrarily large abscissas lying above the line y = -ex. 2. First suppose that lim 'PI (x) = +00, 11:++""

and form the smallest convex domain containing all the points of the curve y = Ci'l{X) and the positive ray of the x-axis. The part of the boundary of this domain that lies above the abscissa-axis is a continuous curve (Fig. I) y = ",{x).

Fig. 1

It is easy to establish the following properties of this curve: (a) the curve is convex from above;13 (b)

lim '" (x) ",++00

x

= 0;

-<

(c) 'PI (x) Hx); (d) each extreme point (that is, not lying inside any line segment of the curve) of the curve y = tp(x) also lies on the curve y = Ci'l(X), that is, at the extreme points Ci'l(X) = tp(x); (e) the curve y = 'f'{x) has a sequence of extreme points that tend to infinity. II

Such functions are called concave.

Sec. 12]

37

PROXIMATE ORDERS

By making inessential changes in the curve y - 'P{x) in the neighborhood of each angular point we may assume that 'P{x) is everywhere differentiable. It follows at once from (a) and (b) that ~' (x)

lim :II

-++00

= o.

(1.54)

Returning to our previous variables, inequality (c) can be written in the form

< '~(ln r), fer) <

In ~ (r) or

rpe+(IIlr)

and finally

f(r>
'" (1Ilr) p+-

Illr.

+ 'P(ln r)/In r we have the inequality fer)

< rP

(r)

(r > 0).

(1.55)

Also, by properties (d) and (e), there exists a sequence of values r .. tending to infinity for which equality holds in (1.55). It remains to verify that the function p(r) that we have constructed is a proximate order, that is, that it satisfies the conditions (1.52) of Valiron. Clearly, from (b) it follows that limr-cop(r) - p. Also, ~

(In r)

rp'(r)lnr =~'(Inr)- -Inr -. Hence, from (b) and (1.54), we see that limr..... aorp'(r) In r - O. We constructed the function per) under the condition that lim sUPZ--+co9'l(X) - 00. 3. We now show how the general case can be reduced .to this case. For this purpose we construct a .concave function y - 'Pl(X) so that

lim ~l (x) a: -+ 00

= 0,

lim 1\I~(x) = 0

a:-+oo·

X

and so that lim sUPz--co[9'l{X) + 'Pl(X)] = 00. We construct the curve 'P1(x) in the following manner: pass a segment dl of the line ~"'-".¥

from the origin to the point Xl (Fig. 2) at which

CP1 (x t ) Choose a positive number EI t1a of the line

>-

'1 X 1

+ 1.

< El and from the point (Xl' -E1Xl ) pass a segment

Y+11 Xt = - li (X-X1)

to a point XI

> Xl' at which

38

GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

[CHAP. I

From the point (x., -E1Xl - Ea(X z - Xl» pass a segment d3 having slope -E3 (0 < E3 < E.) etc. y

r,

-R~~T---~~------~-~

o

r

: \y=rp,(r) '--"-~~-~--y=~(r)

Fig. 2

E3

The positive numbers El , E., E3' •.. , E.. , ••• will be chosen so that El > E. > > ... > E.. > ... and t: .. ~ 0, and the points Xl. Xa, ... , x" • ••• are chosen

so that x" -+- 00. Clearly the polygonal function y structed satisfies the condition

= 1Jit(X) that we have con-

~1 (x) = O.

lim

III -++00 • X

By changing the function tPl(X) in an inessential manner in a neighborhood of each angular point, we can make it everywhere differentiable. Let tpl(X) denote the function with the opposite sign. This function y = tpl(X) has the required properties. Now construct a convex majorant tpa(x) for the function IPl(X) + tpl(X), just as in § 2. Then putting ~ (x)

we obtain

=- ~g (x) - '\"1 (x).

'1

(x) .(: .~ (x)

and, on some sequence {x~} of extreme points tending to infinity, we have

'Pi (x~) = ~ (x~) (n = 1. 2.... ). Also lim oV (x) = 0 and lim III -++00

III -+ +00

0/ (x) = O.

(1.56)

x

Putting, just as before, (Inr) ( ) + 1\11iiT'

p r =p

we obtain from (1.56) lim p(r)=p. r-+oo

11m rp' (r) In r r-++oo

=

lim r-++oo

[.V (In r) _

that is, p(r) is a proximate order. Also fer)

~ rP(r)

r)] =

1\1 (lIn nr

0,

Sec. 12]

39

PROXIMATE ORDERS

and for some sequence r" tending to infinity I (rfI) = r~ (,.fI) • The theorem is proved. Note that the function exp (tp(ln r» is a slowly increasing function L(r). In Chapter II we shall need the following remark. REMARK. The slowly increasing function that we have constructed can be represented, as is easily seen, in the form

L (r) =

,"'.(1:1 ,.)-+.

(III ,.),

where tpl(X) and tpa(x) are unbounded increasing concave functions satisfying condition (c). We shall show that in the expression for L(r) the functions tpl(X) and tp.(x) can be replaced by two other concave functions t'JI(x) and t'J'I,(x) that also satisfy the conditions (a)

Jim 4). (x) =

00,

e-++co

(b)

11m

&t (x)

11-++00

x

= 0,

(I

=

I, 2)

and that satisfy, in addition, the condition

(b')

&r = o.

lim (x) al-+ +co &i (x)

To prove this we make the following construction. Let y = tp(x) be a concave function satisfying the conditions Iims -++ ao tp(x) = 00, Iimz -++ ao tp(x)/x = 0, and let (10) be a line of supportU of this curve. On the line (10) choose a point (xo, Yo) and consider the curve (II)

y

= C~l)+C~l)(X_ x o) - c~1)e-·1 (al-IIIo>,

that is tangent to the line (10) at the point (xo, Yo). In the equation for this curve, £1 is a positive number, and the coefficients c~1), 41) are determined by ~e condition that (10) and (/J be tangent at the point (Xo, yo>

C~l) = .!. (y~ _ C~l~. al

C~l) = Yo

+ .!.. ~ _ C~l~. al

where y~ is the slope of the line (10). The parameter c\l) will be chosen to be positive, but less than y~. Then the coefficient 41) will also be positive. The curve (/J is clearly concave and approaches asymptotically the line y = 41) + c\l)(X - xo). It A line of support of a given curve is a line having points in common with the curve, but .tach that the entire curve lies in one of the closed half-planes determined by the line. (For

.6Irther details on lines of support, see § 19.)

40

GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

[CHAP. I

Also, from the equation

1" (x) y' (x)

=-

+ c~t)e"

C~t)ll

IID-:I:o)

it follows that on the whole curve (/1)

I I< y" (x)

y' (x)

(1.57)

21 •

If the abscissa Xo is sufficiently large, then that part of the curve 11 that lies to the right of Xo is above the curve Y = tp(x). Choosing the curve (/1) in this manner and then decreasing ell!) while keeping the point (xo, Yo) and the quantity El fixed. we can cause this curve to touch the curve y = tp(x) from above (Fig. 3). Since the curve (/1) is concave and contains no line segments, the point of contact (x~. y~) must be an extreme point of the curve y = tp(x). Now choose numbers E2 < E} and ci2) < cP) and choose a point (Xl' Y1) on the curve (/t) far enough out so that the part of the curve (/2) y = C~2)

+- ci2)(X _

Xl) _

c~2;C-'" (.lI-.lI,),

(this curve is tangent to the curve (/1) at the point (Xl' Y1»' lying to the right of this point, lies above the curve Y = tp(x). Without changing the point (Xl' Y1) or the parameter t 2• we decrease the parameter C~2) until the curve (/2) has a point of contact with the curve Y = tp(x). Next we choose numbers E3 and ci3 ) < c\2)

(!.J

Fig. 3

and a point (XI' yJ on I., and form a curve (/J, etc. This process is continued indefinitely, and the numbers E1 > Ell > E3 > ... , cil) > c\1I) > ciS) > ... are chosen so that E" -+ 0 and cl") -+ O. Now we form a smooth concave curve y = ,<x) from the segments of the curves (10)' (/J, ... , (/,,), ... taken between the points of contact. Clearly lim

:z:-+ + co

0 (x) =

00,

lim .lI-++co

& (x) =0 X

and ",<x) ;> tp(x), with equality holding for a sequence of values xi, x~, ... , x~, ... , x~ - co, corresponding to extreme points of the curve y = tp(x). In addition, ",<x) is everywhere twice differentiable, except at the points of contact.

Sec. 13]

EXTENSION OF THE CLASSICAL THEOREMS TO PROXIMATE ORDERS

41

At these points .of contact the first derivative y' = 'I?'(X) has angular points. By changing the function O(x) in an inessential manner these angles can be smoothed out so that the second derivative 'I?"(x) in each interval lies betweel). the upper and lower limits of the function 'I?"(x) at the corresponding points of contact. On the basis of inequality (1.57) we can assert that •

hm z-+ + 00

&" (x) 111 (

v

)

X

= O.

Now choose a majorant 'l?1(X) for the concave function 1f'1(X). Let tii.(x) be the smallest convex majorant for the function CPI(X) = cp(x) + 'l?1(X), and let 'l?z(x) be a majorant for tiiz(x). Then we have everywhere 'l?z{x) tii.(x) 'P(x) + "'l(X), while on some sequence of points x~, x~, ... , x~, ... tending to infinity and corresponding to extreme points of the curve y = Vi(x) we have the equality'l?2(x:J = tiir,(x~). On the other hand, since at the extreme points of the curve y = tii2(X) the equality tiiz(x) = cp(x) + 'l?1(X) holds, we have

>

>

&2(X~) = rp(x~)+ n1 (x~).

Thus we have constructed functions 'l?1(X) and 'l?z(x) satisfying conditions (a), (b), (b ' ) and such that, in addition,

rp (x)

-< 62(x) -

rp (x~) = &2 (x:) - &1 (x~)

&1 (x).

(x: -+ 00).

In the future L *(r) will denote a function having a representation In Le (r)

= 1)2 (In r) -

&1 (In r).

in which the functions "'1(X) and 'l?2(X) are concave functions satisfying conditions (a), (b), (b'). By p*(r) we mean the function defined by rpe(r)

that is, e( )

p r

=(i

= rPLe(r).

+ &2

(In r) - &, (In r) lnr •

The functions p*(r) clearly form a smaller class than the proximate orders p(r); however, as we have seen, this class is sufficient to construct a growth scale for the functions of finite order. More precisely, in the statement of Theorem 16 the proximate order can be replaced by p*(r). The function p*(r) will be called a strong proximate order. 13. ExteasioD of the Classical Theorems to Proximate Orders The classical theorems that were presented in the preceding sections on the CIOnnection between the growth of the function

In M,(r)

42

GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

[CHAP. I

and the distribution of the zeros of the function fez) can be extended to the more precise description of the growth that is given by the proximate orders. The theorem relating the type of the function to the rate of decrease of the coefficients in the Taylor expansion can also be generalized. To formulate this theorem precisely, we introduce the function !p(/), defined to be the unique (for I > 10> solution of the equation t = rP (r). THEOREM

2'.

The type (I, of Ihe entire funclion 00

f (z) =

l: c"r' ,,-0

> 0)14& is git.V!n by the equation

with the proximate order p(r) (p

I

lim cp (n) "-+00 PROOF.

VI e,,1 = (o,?ef',

(1.06')

We first show that lim ,(kt) = k

, .. 00 ,(I)

Indeed, differentiating the equation In t

~

(1.58)

= p(r)1n r, with respect to In e, we obtain

:::: ==p; (r)r In r +p(r) and consequently .

dint

11m d-I ==p.

r-+oo

nr

This equation can also be written in the form lim d In , (t) _ '-+co dlnl -

.!.. p'

Thus we have the asymptotic inequality

(*-.)

d In t

< din cp (t) < (++ I) d In t.

Integrating from I to kt yields-the asymptotic inequality

(~ -.) In k < In ~ ~~) <

(+ + a) In

k.

from which we have at once lim ,(kt) == '-+co ,(I)

leT

14& When P - 0 or p - co, the connection between the rate of arowth of the function aDd the rate of decrease of the coefficients in its power series is rath!z complicated. 1beR have been many papers on this problem; we mention Fridman [I; 1] aDd 8emDeta [I; 1). The inveatiptions of Videnakii [I] and Evgrafov [I] belong to the same c:in:Ie of ideas. We also mention inveatiptions of entire fun<:tions of order zero by BratiiCev and KorobeiDik [I] and by Rubel and Taylor [I).

Sec. 13]

EXTENSION OF THE CLASSICAL THEOREMS TO PROXIMATE ORDERS

43

The remainder of the proof of Theorem 2' closely follows the proof of Theorem 2. From inequality (1.07) we have asymptotically, for a > aI'

In ICn I < arP (r) -

n In r.

Choosing r to be the root of the equation

n=

aprp(r)

we have

In Icn I or In (?

<;-

n In cp

(;a ),

(.iv 1c. I) < : +10 :G,) ,

and passing to the limit we have

lim (cp (n) n~co

and, since a

YIc

n

1

p< (apefp •

> a, was arbitrary, (1.09')

We must show that equality holds in (1.09'). We define a by the equation n

lim cp (n) n~co

VIc:i =

1

(ape)p

and show that the assumption a < a, leads to a contradiction. Choose any number 0"1 between a and at (a < a1 < a,). We have for all sufficiently large n 1

IC I < { (alpe) P n

cp (n)

}n.

Using (1.58) this asymptotic inequality can be put into the form 1

Thus. for all n

> "0. eP

I cnr"" I < { (~) cp

alP

}IIrn

44

GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

[CHAP. I

and the maximal term I-',(r) of the Maclaurin series of j(r) satisfies the inequality

(:) r" I

,.,<,) < ~I

If

alP

The maximum of the right-hand side is attained for n = [a, p,.,,
< eG,rp(r).

Hence from (1.10) we see that for all PI

M f (r)

>P

< (2 + 2 'eaj hrP,) eo,rP Ir), P

and this leads to the contradiction (1, ~ (11' Thus the inequality sign in (1.09') cannot occur, and the theorem is proved. We pass now to the theorem on the relation between the type of an entire function with a given proximate order and the distribution of its zeros. Let j(z) be an entire function with proximate orde; p(r), that is, -:- In Mf(r)

af =

Then for

E

lim

r~

=I=-

0, co.

(1.59)

rpI')

00

> 0 we have asymptotically In Mf{r)

and by Lemma 4 nf(r)

< (a, + ; )rw ),

< (a, + ; )(ert ler ),

Using the asymptotic inequality (see (2) following Lemma 5, § 12) (erl(er)

=

ePrPL (er)

which is valid for arbitrary 7J

< (I + "I) ePrPL(r) =

(1

+ "I)

ePrp(r).

> 0, we have from (1.59) -.- n,(r)

11m

-IT)

r~oo r P

-< ePa"

Retaining the notation -.- n,lr)

6.,= r h m-.... oo rP IT) , we write this inequality in the form

6.,-< ePa,. If P is not an integer we can obtain a converse estimate. In this case, from the representation (\.38) of the entire function and from Lemma 3, we have

( rn dt f n~P+idt) In M,(r) -< 0 (r")+ kllr" r ~1I+1 + r r

co

(I)

(I)

r

Sec. 13]

EXTENSION OF THE CLASSICAL THEOREMS TO PROXIMATE ORDERS

45

From the asymptotic inequality

n,(r)

< (A, + l) rP(r,

and property (3) of proximate orders we have InM,(r)<:kp(A,+a)[p 1 p

+p+~_p]rp(r'+O(rp(rl),

from which follows

a,<: cptl" where cp is a constant not depending on the functionj(z). Thus we have obtained a theorem that generalizes Theorem 14. THEOREM 17. Let j(z) he an entire function of nonintegral order p and let p(r) be a proximate order ....ith lim per) = p. If with this exponent = 0, then fez) is of minimal type ....ith respect to per), i/O < < 00, then of normal type, = 00, then f is of maximal type. and if

at

at

at

The case of integral order is exceptional here too. We have already seen in § 10 that in this case the value of the upper density is not enough by itself to determine whether the function is of minimal, normal or maximal type. We define the quantity {Jt for entire functions of order per). Whether or not the series

at

converges, we represent the entire functionf(z) of integral order p in the form

...

!(Z)=zme where P(z) = (XpzP

P(II,!!a(:n; p)

(01<:00),

+ ocp_1z p- 1 + ... + otlZ + CXo. 3,(r) = a p +..!.. p

(1.38')

We put

~ _1-

~ aP lanl
and

3,= 11m 16,(r) I r .. co L (r)

,

_here L(r) = rP(')-p, and we introduce the quantity

1, = max (A"

a,).

The following generalization of Lindelof's theorem answers the question of the type of a function fez) with respect to a given proximate order. THEOREM

18.

Let fez) be an entire function of integral order p and let p(r)

.{te a proximate order with limr -+ ao p(r) = p > I. ., If Yt = 0, then fez) is of minimal type with respect i}.enf(z) is of normal type, and ifYI =

00,

to p(r); if 0 < Yt thenf(z) is of maximal type.

< 00,

46

GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

[CHAP. I

We give the proof briefly, using the estimates that we found in the proof of Theorem 15(a). The functionf(z) can be represented in the form (l.40) whether the genus of the canonical product coincides with the order of the function or is less than it. From the bound (1.41), (1.53) and (1.53'), it follows easily that asymptotically

(1.42') Dividing the asymptotic inequality (1.43) by limit as R -+ 00 we have

RP{rj

and taking the upper

(1.44') To obtain the reverse inequality we note that from the estimate nCr) <; In M(er) we have the inequality Il, ePa, (1.45')

<-

for the proximate order. Also, from inequality (1.46) we have, using (1.42') and (1.45'),

0, <. cIa" where

C1

is a constant. From this last inequality and (1.45') we have:

-r,.

ar:? c- 1

( 1.49')

The assertion of the theorem follows from (1.44') and (1.49'). The concept of the category of a function with respect to a given proximate order can be introduced, just as in § 9 for the usual order. Two functionsf(z) and q:(z) are said to be of the same category if O"t and 0"" are both infinite, or both finite and positive, or both equal to zero. Theorem 12 can be extended to these generalized categories. namely: The category of the product of two functions fez) and cp(z) does not exceed the larger of the categories of the factors, and if the categories offez) and cp(z) are different, then it coincides ....ith the larger of these categories. The proof can be given just as in § 9. However, it is simpler to reason as follows, using Theorems 17 and 18. From the equation n,.(r) = nt(r)

we have

At, -<- At

+ n,(r)

+ ll.,..

while if either of the terms on the right side is zero or infinity, then equality holds. For functions of nonintegral order the quantities 0", and 6, are either both zero, or both finite and positive, or both infinite. The assertion of the theorem follows at once for this case. For integral p = lim,,_ao p(r) it is necessary to

Sec. 14]

THE PHRAGMEN-LiNDELOF PRINCIPLE

47

consider the relation

o,,(r) = o,(r)+o,(r). The assertion of the theorem for integral p can be obtained easily from these relations if we note that by Theorem 18 the category ofa function is completely determined by the quantity y, = max (15" til)' The corollaries to Theorem 12 can also be carried over to the case of generalized categories. The crude characterization of the growth of a function that is given by the asymptotic behavior of M,(r) is insufficient for a deeper study of the relation between the growth of the function and the distribution of its zeros. In § IS we shall introduce and study a more precise growth characteristic, in which the growth of the function is studied in its dependence on the direction in which z tends to infinity. It will then turn out that the growth of the function depends in an essential manner on the distribution of the arguments of the zeros, even for the case of non integral order. To study this growth characteristic we need some general theorems of the theory of functions which we shall present in the following sections. 14. lbe Phragmen-Lindeliif Principle The maximum principle states that a function j(z) that is holomorphic in a domain and continuous on the closure of the domain attains the maximum value of its modulus on the boundary. This important principle was extended by Ph ragmen and Lindelof [1] to the case when the function is not continuous at certain exceptional boundary points, under the assumption, however, that as these points are approached the modulus of the function does not grow too fast. The following theorem is fundamental. THEOREM 19. Let f(z) be holomorphic in a domain G, and suppose there exists a holomorphic function w(z) in this domain such that f(z)[w(z»)" has, for all ~ > 0, a limit at each boundary point of G (including the point at infinity, ([ il is a boundary point), and that on the entire boundary

I f(z) (w(Z»)3 1-< M, where M does not depend on Then throughout the domain G.

(1.60)

~.

If(z) I

<M

PROOF. From the usual maximum principle it follows that if (1.60) holds on the boundary of G, then it also holds throughout the interior of the domain. 15 ,

11 If the function w(z) has roots in the domain G, then f(z)[w(z)l" is not holomorphic in G, but the modulus of this multiple-valued function is single-valued in G. The maximum modulus principle can be extended to this case without essential modification of the proof.

48

GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

[CHAP. I

Let the point Zo belong to the domain G and let w(zo) :;1= O. Then for arbitrary

15>0 and consequently

(1.61) I/(zO>I~M. The roots of the function w(z) are isolated points and, by the maximum principle inequality (1.61), must continue to hold at these points, that is, throughout the domain G. Note that equality cannot be attained at any interior point, unless j(z) is a constant. From this general principle we obtain a number of important tJ!.eorems that are frequently applied in various questions. THEOREM 20. Let f(z) be holomorphic in the domain G, whose boundary contains the point at infinity, and suppose that at all finite boundary points

I/(z) I <: M.

1f /f(z)/

(1.62)

is bounded in G, then inequality (1.62) holds throughout G.

PROOF. Remove from the domain G a disc C of radius r with center at some point zoo Put r w(z)=--. Z-Zo

The function f(z)[w(z)]" tends to zero at infinity, and since /w(z)/ /z - zo/ > r, we will have on the boundary of G

<1

for

II (z)[w (Z»)8 I ~ M. In addition, on the circumference /z - zol = r, we have

II (z)(w (Z))8 I ~ M1, where Ml = maxlz-zol=r /f(z)/. By the general Phragmen-Lindelofprinciple we have, throughout the domain formed by removing the disc C,

I f(z) I ~ max(M,

M 1)

~

EO-C).

Now let r tend to zero. Then lim r_ o Ml = I/(zo)I and

I/(z) I ~ max (M, If If(zo)1

II (zo) I>

(z EO).

> M, then we would have throughout G

II (z) I~ II (zo) I· But at the interior point Zo the function I/(zo)I cannot attain a maximum. Therefore If(zo)1 < M and consequently I/(z) I ~ M for z E O.

Sec. 14]

49

THE PHRAGMEN-L1NDELOF PRINCIPLE

The theorem on the growth of a function inside an angle will be very important for us. This theorem was obtained by Ph ragmen and Lindelof from the general principle. First we introduce the following concepts. Let M,(r.

II.

~) =

and define the order P and the type Ot .;;; arg z .;;; fJ to be the quantities _

p= lim

r-.co

In+ln+M,
(/f

p)

nr

max

.<.<~

If(rei ') I

of a function fez) analytic in the angle

and

a,=

_

Um

In + M,(r, a,

r-.':".

p)

r'

•

THEOREM 21. Let the functionf(z) be holomorphic inside an angle of opening TT/Ot and continuous on the boundary. Assume that on the sides of the angle

If(z)I~M

and that the order P of the function is less than Ot. Then If(z) I ~M

throughout the angle. PROOF. Without loss of generality we can consider the angle larg zl <; TT/2Ot. Let

where P < y

< Ot.

Then asymptotically inside the angle larg zl .;;; TT/2

If (z)[w(Z)]3 I = and consequently for p

If(z) I ,-&rTcoet'

(z=re".

a> 0)

< PI < y If (z)[w (z)]31 < ,rP. -arT 008 y8.

Thus we have shown that the function

f(z)[W(z»)3 tends uniformly to zero inside the angle larg zl <; TT/2Ot as r -+- 00. Further, since the modulus of this function is bounded by M on the sides of the angle. ·we have by the general Phragmen-Lindelof principle

,

If(z) I ~ M

throughout the angie. The theorem is sharp in the sense that there are functions of order p inside tin angie of opening TTl p that are bounded on the sides of the angle. An example ,or such a function is ,rI.

SO

GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

[CHAP. I

taken inside the angle larg zl '" 'TT/2p. However, for the case of an angle of opening 'TT/ p, one can give an estimate of the modulus of the function inside the angle that depends on the type. 22. If fez) is a function of order p and type (J ;> 0, holomorphic inside the angle larg zl ~ 'TT/2p, and if on the sides of this angle THEOREM

then throughout the angle we have the inequality

!! (rei') I <. Meer'

008

p.

(1 61<.i)· I.

The angle larg zl " 'TT/2p can, of course, be replaced by any other angle of opening

'TTlp· PROOF.

The function

IP. (z) =

r(crH)zP

f (z)

is bounded on the sides of the angle larg zl <; 'TT/2p and, in addition, is bounded on the positive real axis. We apply Theorem 21 to this function inside each of the angles 0 <; arg z <; 'TT'/2p and -'TT/2p <; arg z '" 0, and thus we see that the function fII.(Z) is bounded throughout the full angle. Hence, by Theorem 20, inside this angle we have

11P.(z)I <. M

or

Ifez) 1<' Me(crH)r P coap. and finally, since. e was arbitrary,

If(z)1 <. MeGrPooap•• In particular, if the functionf(z) is of order 1 and type plane (1m z ;> 0) and is bounded on the real axis

If(x>I<'M

(-00

(J

in the upper half-

<x
then throughout the half-plane 1m z ;> 0

If(z) I <. Me"lma.

(1.63)

For (J = 0 Theorem 22 can be regarded as a strengthening of Theorem 21. Indeed, in this case, it differs from Theorem 21 in that the requirement that the order p of the function.f(z) inside the angle be less than a is replaced by the weaker requirement that the function be at most of minimal type with respect to the order p = a inside the angle.

Sec. 15]

51

THE INDICATOR FUNCTION

COROLLARY. If the entire function f(z) is at most offirst order and minimal type and if its modulus is bounded on some- line, then it is a constant.

Indeed, in this case, 1/(z)1 is bounded in each of the half-planes determined by the line, that is, 1/(z)1 is bounded in the entire plane, and hence, by Liouville's theorem, it is a constant. We note that by completely analogous reasoning we can establish the following Phragmen-Lindelof theorem for the strip. Letf(z) be analytic in the strip Iyl ~ 1', with

I f(X±ll)1 < M and If(x+l,v)1 < Nee'er'" I (M> 0 and N> 0) and 0 ~ k < 7T/2y. Then 1/(z)1 ~ M throughout the strip. Indeed, choosing w(z)= e-e"ll

we have

1w (z) 18 = e- 8e":':

008011

(3) 0).

Putting MI = max 1/(iy)1 for -I' ~ Y .;; I' and Ma by the maximum modulus principle,

=

max (M, M 1) we have,

If(z)(w(z)}81 ~ M2 in the half-strip x;> 0, Iyl ~ y. By the general Phragmen-Lindelof principle we then have 1/(z)1 .;; M'l. for x ;> 0, Iyl .;; y. In the same way one proves this inequality for the left half strip x .;; 0, Iyl ~ y. Now from Theorem 20 we have If(z) I

<M

for Iyl .;; y. 15. De Indicator Function To characterize the dependence of the growth of a function f(z) of finite order p in an angle 01 ~ arg z ~ Os on the direction in which z tends to infinity, Ph ragmen and Lindelof [1) introduced the function h,(IIII.) =

11·m In "(re") I r"CIO

(II.II, ~ / 6~ /6II' \

(1.64)

"

called the indicator function of fez). Note that for functions of normal type inside an angle the indicator is bounded above, and for entire functions of normal typel8 it is bounded below because of Theorem 11. lt follows at once from this definition that the indicator of the product of two functions is less than or equal to the sum of the indicators of the factors h,,(8)~h,(6)+h/(fj) (1.65) II

Tr_lato,'s IfOte: We assume f"

o.

52

GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

[CHAP. I

and the indicator of the sum of two functions is less than or equal to, for each value of 0, the larger of the two indicators

h,.f, (6)

-< max (h,(f:I).

(1.66)

h, (6».

Let us verify the second of these assertions:

It is easy to see that if, for some value of 0, the indicators of the two summands are not equal, then equality holds in (1.66). For the function

f(z) = holomorphic in any angle

Cl ~

0

~

e(a-ib)~P.

{J, {3 -

If (ret~) I =

(l

~ 277, we have

e(a cos p9+b sin pe),.p

and its indicator is equal to

a cos p6 +b sin p6. Such an indicator will be called "trigonometric," and the expression a cos pO + b sin pO will be denoted by H(O). It is easy to see that the function H(O) that takes the values hi and h2 at the points 0 1 and O2 is given by the formula

If (6) = hi Sin p (83 - 8)

+ ha sin p (8 -

sin p (8a-

8t )

8t>

(1.67)

•

LEMMA 6. Let h(O) be the indicator for a function of order p, holomorphic in the angle 01 ~ 0 ~ 2 , O2 - 01 < 77/ p, and

°

h,(fJ 1)

-< hI'

h,(6~

-< h'J;

let the function H(O) be defined by (1.67). Then for all 0 in the interval (° 1, h (6)

-< If (6).

( 2)

(1.68)

This property is called trigonometric convexity. PROOF.

Construct the function

that takes the values hI (tJ 0). The function

>

If. (6) =

a.

cos p6

+ tJ and

h2

+ tJ

+ b. sin p6.

at the points

()l

and

()2,

respectively

Sec. 16]

53

ANALYTIC PROPERTIES OF THE INDICATOR FUNCTION

has the indicator 17

= h,(6) -

Ha (6). Thus, along the rays arg z = 1, 0z, the function !p(z) tends to zero. By the Phragmen-Lindelof theorem tp(z) is bounded throughout the angle 01 <; arg z <; 0z and consequently its indicator i~ less than or equal to zero. Thus for all !5 > 0 h,(IJ)

°

h,(6)

< He (6),

61

< 6 < 62,

° °

from which (1.68) follows. It follows from Lemma 6 that if the indicator function is finite for = 01,0,. (Oz - 01 < TTl p), then it is bounded above for all values of in the interval 01 <; <; Oz. Further, if for some value = 00 in this interval

°

°

h(6o)=-oo , then h(O) = - 00 in the whole interval (° 1, OJ. This last is easy to obtain if we take 00 as one end point of the interval and change the reasoning of Lemma 6 slightly. Sometimes the following equivalent formulation of Lemma 6 is convenient: If 01 < O2 < Os, Oa - 01 < TTl P and h(Ol) <; H(Ol)' h(02) ;> H(OJ, then

h (6 s)

>- H (6s).

It follows from this formulation that if h(Oz) some angle containing the ray = Oz.

°

=

00,

then h(O)

=

00

inside

16. The Fundamental Relations and Analytic Properties of the Indicator Function Using equation (1.67) we can reformulate Lemma 6 on the trigonometric convexity of the indicator in the following symmetric form. p

THEOREM 23. The indicator h,(O) of a function f(z) holomorphic and of order inside the angle 01 ~ arg z ~ Os, Oa - 01 < TTl p, satisfies the relation

h (6 1) sin p (6 9 -

lor 01

63)

< O2 < Oa.

+ h (62) sin p (6a -

( 1)

+ h (6s) sin p(6

l -

6;)

<0

(1.69)

°

Inequality (1.69) will be called the fundamental relation of the indicator. . This inequality is unchanged by cyclic permutation of the quantities 1, 0., Os and changes sign if any two of these quantities are interchanged. From the fundamental relation we obtain a number of analytical properties of trigonometrically convex functions in a similar manner to the way in which they are obtained for ordinary convex functions. " Here we are using the following general fact: if Iim,_ '" 'P
11m [fer)

r-+co

±, (r)] = r 11m fer) ± 11m ,(r). .... oo r .... oo

54

GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

[CHAP. I

For this we first transform the fundamental relation as follows: h (1:1) sin p (6s -

( 1)

+ h (6

+ h (1'8) sin p (61 -

1)

(sin p(61 -

6) 63) - sin p ("1 -0))

-6J 6-6 6,-6 <. 4h (6 1) sin p -6a 2 - sin p - 2 - slnp -2- (6 1 < 6« t

3),

(1.10)

Dividing both parts of the inequality by sin p«() - ()l) sin p«()s - ()l) we obtain h (0) - h (Ot) ~ k (Oa) -

sIn p (6 - 01)

"'"

h (0 1)

sin p (01 - 01)

Os-Ot 6-6 + h (61) sin p -°1-0 2 - sec p - 2 - sec p -2-· . t

(1.11)

Transposing the quantities ()l and 0, and taking account of the sign of the denominator, it can be shown that inequality (1.71) holds also for () < ()l < ()s. For ()3 < ()l < 0 and for ()s < () < ()l the sign in inequality (1.71) is reversed. We now establish the following property of the trigonometrically convex function h«() that is bounded on some interval: (a) The function h«() is continuous. It is clearly sufficient to prove this for intervals of length less than TTI p. Considering (1.71) for ()l < () < ()3 and for ()s < ()l < (), we see that for () > ()l the ratio h (0) - h (Ot) sin p (0 - 61)

is bounded. Considering (1.71) in the other two cases we see that this ratio is also bounded for () < ()l' From the boundedness of this ratio the continuity of h«() clearly follows. (b) h«() has a derivative from the left and from the right at every point. PROOF.

Let k

for I()s - ()ll <; q obtain

< 7Tlp

=

max

and ()l

It h(6)sec2 pfl

< () < ()s.

From inequality (1.71) one can

h(O)-h(Ot) ~1t(0s)-h(Ot)+k(6 -6 )-(0-6 »), sin p (0 - 0t> "'" Sin p (Os - 01) 8 1 1

(1.12)

and consequently the bounded quantity

r(6 •

6-'= h(O).-h(Ot> +k(6-6) 1J sin p (0 - 0t> 1

decreases monotonically as () -+ ()l limit (61) =

h:

+ O.

From this follows the existence of the

11m (pr (6, ( 1)],

....... +0

Similarly one shows the existence of the derivative from the left.

Sec. 16

55

ANALYTIC PROPERTIES OF THE INDICATOR FUNCTION

(c) The right hand derimtive is greater than or equal to the left hand derh'ath'e at each point:

This inequality is immediate if one writes (1.71) with 0 < 01 < Oa and passes to the limit as 0 -+- 01 - 0 and Oa -+- 01 + O. (d) The derivative h'+(O) is continuous from the right and the derivative h'-(O) is continuous from the left. PROOF.

Choose 01

< 0 < Oa.

It (0) - It (08) sin p (0 - 08 ) - h (83 ) :p. hsin(01) p (81 - 6s)

-

Just as we obtained (1.71) we can obtain

8- 6 2

.

h (!:IS> sin p - -1 sec p

61 - 88 8 - 88 sec p - 2 2

or It (8) - It (8a) ~ It (Ot> - h (fls) sin p (6 - 8S> ~ sin p (8 1 - 68 )

k(6 -01),

(1.73)

Passing to the limit as 0 -+- Oa - 0 we obtain h~

(as> ~ pr (6t •

6/1) -2pk( 03

-

( 1),

Similarly to (1.72) we obtain h'+(OI) < pr(OI' ( 3 ), It follows from the last two inequalities that

h~ (.6 3) - h~ (fjl) ~ - 2kp (I}s - ( 1);

(1.74)

and comparing this with property (c) we see that the functions

h~ (fJ)+ 2kpO and h~ (6)

+ 2kpO

are nondecreasing on the interval [01, 01 + q]. Thus the limits h'-(O - 0) and + 0) both exist. Passing to the limit as s -+- +0 in the equations

h~(O

f

8+.

h(8+s)-It(0)

s

=..!.. s

and

h~(Ij)dfj

+I h~ 8

h (8 -

~ ~ h (0)

=

(6) d6.

8-e

we obtain

"

h+ (6) = h+ (6

+ 0)

and h_ " (6) = h_ (6 - 0) .

(e) The function h(O) has a derivative at all points except possibly on a countable set.

S6

GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

[CHAP. I

For the proof we write (1.74) in the form I

I

h+ (61) -

h_ (61) -

-< h_ (6.) I

2kp (6 s -

(1)

I

h_ (61) •

If 01 is a point of continuity for h'-(O), then I

h+ (a l )

I

<: h_ (01) •

Comparing with property (c) we see that h(O) has a derivative at all points of continuity of the monotone function h'-(O) + 2kpO, that is, at all points except possibly on a countable set. (f) If 00 is a local maximum or minimum point for the function h(O), then

for 19 - 90 1<; 'TTl p

>

h (6) h (a o) cos p (6 - 60>. Indeed, at a maximum point h'(Oo) = 0, while at a minimum h~(Oo) ;;> 0 ;;> (h'-0o). Putting 01 = 00 in (1.71) and passing to the limit as 0-+00 + 0, we obtain h (Os) - h (00)

sin p (Os - 00)

+ h (0)' Os - 0 0 sm p 2

0

sec p

Os -- On ........

2

-:?

.!.. h'

(" ) ........ 0 + Vo -:?

P

or for 0 < 03 - 00 < 'TTlp. Similarly one obtains an estimate to the left of the point 00 , Note that if 00 is a local maximum for h(O) and if h(Oo) = 0, then, by what has been shown, h(O) == 0 in some neighborhood of the point 00 , (g) For arbitrary IX and fJ (IX

< fJ)

(1.75)

•

and equality holds only for h (0) = a cos pO

+ b si n pO

(ex

-< 0 -< ~).

where a and h are constants. PROOF.

From inequality (1.71) it follows that as 0 -+ 01 :

(0 ) ~ h (Os) - h (°1) + 1 ...." sin p (Os ~ 0t>

2... h' P

+ h (0 ) tg p Os -2 °1 • 1

Interchanging 01 and 03 we have

2... h' p

-

(0» J

h(OIl)- h(Ot) -h(as)tgp 611 -;6 1 sin p (0, - 61)

and, after subtracting these inequalities,

1 I I 0l)]+[h(61) p[h_(Oa)-h+(

+ h(6).]tgp-2-> 6a -O 0. l

(1. 76)

Sec. 16]

57

ANALYTIC PROPERTIES OF THE INDICATOR FUNCTION

Subdivide the interval (rx, (J) by means of points 01 , O2 , ••• , 0n_l at which the derivative h'(O) exists; write inequality (1.76) for each of these subintervals and sum the results. We obtain _1 I

~

I

h_O)-h+(a)+p ~(h(6j)+h(6J+l)ltgp

8i + l

-

2

8 i

>0

j=O

=

60 ,

~

=

6n ). To obtain inequality (1.75) it is necessary to pass to the limit as max 18;0'-1 - 0;1- o. Suppose that for some rx and {J ({J > rx)

(a

sea, ~)=h~OO)-h~ (a)+pll

Since for rx

< 0 < (3

r II

J

h(/j)d6 =0.

II

I

I

sea, ~)=s(lJ, /j)+s(6,~)+h+(6)-h_(6)

and we have

I h_(&)+pll

f•h(&)d6=h+(a), I

II

or

+

h" (6) p2h (6) = O. Thus equality holds in (1.75) only for h (&) = a cos pO

+ IJ sir. p6.

Property (g) is characteristic for trigonometrically convex functions. In other words, the necessary and sufficient condition that the function h(O) be trigonometrically convex for some p on the interval (rx, (J) is that the function

f•

5(6) = h'(6)+p'l h(cp)dcp be nondecreasing on that interval. It is only necessary to prove the sufficiency. First assume that s(O) is a differentiable function. Then h" (&)

+ p'lh (6) =

(1.17)

5' (6).

(1.17') The Green's for the differential operator h" + on the interval 01 < 0 < 83 (0 < 01 < 03 < 17/ p), with the boundary conditions G(OI' 8) = G(03' 0) = 0, has the form sin p (811 - 8) sin p(.} - 6t ) for p 2h

function 18

0(4,6)=

11

I

4 < 6,

p sin p (81 _ 8s )

sin p (8 - 6t ) sin p (O~ p sin p (81 - 8s)

4)

For the Green's function see, for example, G. Sansone [1].

for '"

> 6.

58

GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

[CHAP. I

The solution of equation (1.77) can be written in the form h (~) =

+

h (6 1) sin p (68 - 8) h (0 3) sin p (6 - 61) sin p (68 - 61) 8.

+ f O(~,

O)ds(~).

(I. 78)

I,

If s(O) is not a differentiable function then we approximate it by nondecreasing differentiable functions and pass to the limit in (1.78), and thus in this case too we obtain a representation of h(O) in the form (I.78). Since G(V', 8) < for 01 < 11', 0 < Os and since s(V') is a nondecreasing function, we have from (1.78)

°

(I. 79) h(O) is a trigonometrically convex function in every interval in which the function h'(O) + p2 h(O) dO is nondecreasing. Equality in (1.79) for even a single value of 0 7fI= 01, Os is only possible if the integral in (1.78) is equal to zero, that is, if s(O) is constant on the interval 01 < V' < 03 , or in other words, if h(O) = A cos pO + B sin pO. Formula (1.78) remains valid for 83 - 01 > 17/ p. It is only necessary that 03 - 01 should not be an integral multiple of 17/ p, since then the Green's function does not exist. We now show that every nondecreasing function S(O) determines a periodic, trigonometrically convex function h(O). For this we must construct the Green's function for the differential operator h W + p 2h with the periodic boundary conditions

r

0(0, fJ)

= 0 (21t,

,

fJ) and O<\, (0,

~)

=

,

O<\, (21t,

~).

For p not an integer the function

o ('f.

~) = 2-7tcos P (fJ - of -'It) P sin 7tp

satisfies the conditions O(O-21t,

,

~)=O(6,~) andO<\,(~-21t.

,

6)-0<\,(0,

~)=1.

Continuing this function periodically with the period 217 we obtain the desired Green's function. If the function s(V') is differentiable, then the function 21<

h (11)

=

faN.

Ij) ds (I})

o is the unique periodic solution of equation (1.77'). If s(V') is nondifferentiable, then we approximate it by differentiable functions and pass to the limit, and thus we see that in this case too this representation is valid. If 0 is a point of

Sec. 16]

S9

ANALYTIC PROPERTIES OF THE INDICATOR FUNCTION

continuity of s(¥'), then the integration from 0 to 21T can be replaced by the integratiort" from 0 - 21T to 0 if we put s(O + 21T) = S(21T) - s(O) + s(O). Then we shall have 8

SCOSP(~-I\I-1t)dS('}).

h(6)=2p s.1n 'ltp

(1.80)

8-2..

Formulas (1.77) and (1.80) are inverse to one another and establish a connection between all periodic, trigonometrically convex functions for a given, nonintegral p, and all nondecreasing functions on the interval (0, 21T). The case of integral p is somewhat more complicated. First we note that in this case if the trigonometrically convex function h(O) with period 21T is given, then the function s(O) defined by (1.77) must satisfy the condition 2..

f

eipo\l ds('}) =

o.

(1.81)

o

Indeed 2K

2..

2..

f eipo\lds(''f) =P~ I eiPo\lh('})d'~+ f eipo\ld{h'('})}, o

0

0

and, transforming the second integral by integrating by parts twice, we obtain (1.81). We now show that to every monotone function s(¥'), satisfying (1.81), there corresponds a trigonometrically convex function of period 21T satisfying equation (1.77). However, this solution of equation (1.77) cannot be constructed as in the case of nonintegral p. Indeed, for p an integer the homogeneous equation h" + p2h = 0 has the periodic solutions cos pO, sin pO, and therefore there is no Green's function satisfying periodic boundary conditions. However, in this case one can construct a generalized Green's function that gives the periodic solution of the inhomogeneous equation (1.77') when s'(O) is orthogonal . to the solutions of the homogeneous equation cos pO, sin pO. To construct this generalized Green's function we note that the function

0t<·~,

6) =

~p

,

(6 -'f) sinp(6 -1\1)

satisfies the equation Also,

O~88 + p20t =

2C05 P (6

-1\1).

0t(6-21t,fJ)-Ol(IJ,6)=0, O~",(fJ-21t, fj)-O;",(fJ,fJ)= 1. If we continue the function G(¥" 8) periodically with period 21T, we obtain the

60

GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

[CHAP. I

generalized Green's function. Thus if s(tp) is a differentiable, nondecreasing function,18a satisfying (1.81), then every periodic solution of (1.77') can be represented in the form B

r

h(fJ)=i(0-4)sinp(fJ-'I)ds('t) +AcospfJ+BsinpfJ. 7tp &-'21<

(1.82)

It remaInS to consider the case of a nondifferentiable monotone function s(8),

satisfying condition (1.81). In this case we must approximate s(8) by differentiable functions satisfying (1.81). For example, for the approximating function we can take B+k

Sk

(0) =

i- J

S

(qJ) d~.

(Here we suppose that s(O + 27T) = s(8) + S(27T} - s(O).) Then one must pass to the limit in formula (1.82). Thus for integral p formulas (1.77) and (1.82) are inverse to one another and establish a correspondence between all periodic and trigonometrically convex (for the given p) functions h(O) and all nondecreasing functions s(O) on the interval (0, 27T) that satisfy condition (1.81). We formulate the result as a separate theorem. THEOREM 24. The general form of periodic functions h(O) with period 27T is given, for non integral p, by the formula 8

h(fJ) = 2 -1 1 p s n 'ltP

f

where s(tp) is an arbitrary nondecreasing function on (8 - 27T :( integral p by the formula

h(fJ)=2!P

(1.80)

cosp(fJ-4-'lt)ds(4).

8-21<

• f (6-4)sinp(6-4)ds(ljI)

tp :(

+Acosp6+Bsinp6.

8), and for

(1.82)

La.

where A and B are constants and s(tp) is an arbitrary nondecreasing function satisfying the condition 2K

I

eip'l' ds (4)

= o.

(1.81)

o ... Here and later a Stiel\jes integral from 8 - 2". to 8 is to be taken as an integral from 8 - 2". - 0 to 8 - O.

Sec. 17]

AUXILIARY FUNCTIONS

61

The function s(8) is expressed in terms of h(8) by the formula 1e s(fJ) = h'(fJ)+pi

f•h(,)d,.

We shall use Theorem 24 in Chapter II. In conclusion we note an almost self-evident property of the indicator, which we shall frequently need: (h)

This property follows at once from the fundamental inequality (1.69) if we put 81 = 0, O2 = + r./2p and 83 = 8 + r./p.

°

17. Auxiliary FUDCtioos In this section we investigate some special entire functions that we shall need in the study of the indicator for proximate orders. These functions will play in the theory of proximate orders the same role as was played by the functions

eAzP in the study of entire functions of order p and normal type.20 In addition, these functions will playa fundamental role in Chapter II in the discovery of asymptotic formulas for functions with very regular distribution of zeros. We shall have to consider logarithms of canonical products, and to choose a single-valued branch we shall define the function

In O(u; p) in the u-plane cut along the positive real axis from 1 to 00, with arg G(u; p) = - 7T along the upper side of the cut. If in the z-plane we introduce cuts along rays emanating from the zeros of the canonical product II(z) (such that the backward extension of each ray would pass through the origin), then in the resulting star domain the function

f

co

In II (z) =

In a

(a:; p)

is defined and single-valued. I I One can verify directly that the functions represented by formulas (1.80) and (1.82) satisfy equation (1.77). This gives an independent proof of Theorem 24, since for p nonintegral the periodic solution of (1.77) is unique, while for p integral it is defined up to the addition of terms of the form A cos pfJ + B sin pfJ. • These functions were first constructed by Val iron [3].

62

(jENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

[CHAP. I

We first give an asymptotic expression for the function In n(.::) in the special case when all the zeros of H(z) lie on one ray, and the set of zeros has a density ~ and p is not an integer. Note that it is not difficult to construct a sequence {Ok} of positive numbers having a given density ~ with respect to a given proximate order p(r).For this it is sufficient to choose ak to be the unique positive root of the equation k = tlrP (r) • To obtain the asymptotic formula we need two lemmas. LEMMA

7.

If the set ofpoints {Ok} is such that for some C > 0 and proximate

order per) n (r)

and

f~ (z) =

< CrP (r)

II a (:k; p)

(p

(1.83)

< p; 0 < a

( 1.84)

<; 1/2),

I ak I <ar

then asymptotically

I In fa (z)1 < C/JP-Prp(r),

(1.85)

where the comtant Cp does not depend on a.2Oa PROOF.

For p

> 0 it follows from

inequality (1.21) that ar

In

I!. (z)1 ~

~

"'"

A rP+l P

r

dn (/) --. tP (t r)

+

o

or

f

~r

Inlf~(z)I
rP+1n(ar) +A (p+l)rp+l p(ar)P (1 +a)r P

o

n(t)dt tP+1(t+r)

< APC aP(ar,(1\P+ a) +AP (p+ I)crp •r tplt) <W

(
p -1 dt.

o

The last integral can be calculated by formula (1.53) and thus asymptotically In I f~ (z)1

< C;ap- PrP (r),

where C; does not depend on a. On the other hand, for a <; 1/2 and luI ;> 1/ a we have InII - ul ;> 0, and consequently InIG(u,p)l;> -Iul-

2

lul Tlul - ... - p > P

-luIP(I

+ In p),

whence

InIG(u,p)l> -Aplul p + 1/ (I + luI). lOa To obtain a singl~valued branch of argf(z) we cut the plane along the ray arg(z/ak) - 0 from ak to infinity, and take argJ..(O) - O.

Sec. 17]

63

AUXILIARY FUNCTIONS

If we use this inequality instead of (1.21), we obtain, by reasoning similar to that used above,

and finally (1.86) For p = 0 and p > 0 inequality (1.86) is obtained similarly from (1.22). Now let us estimate argfa(z). We have

I arg a (u;

" I: < 21t + ~ I ~I < A; Iu I"

p)1

for

I u I> 1

k=1

from which we obtain

".,

I arg to (z)1

< A;rp Sd;~t) •

o Estimating this integral as we did the preceding one, we have

I arg to (z)1 < C;ap-"rp (r).

(1.88)

The assertion of the lemma follows from (1.86) and (1.88). LEMMA 8.

Let the sequence {ak } satisfy (1.83) and let

II 0(:; p)

~/(z) =

I (Jill >ow

(for

p<

or> 2.

p+ 1).

II

Then asymptotically

lin t/(z)1 where the constant PROOF.

C(p)

< C(p) 'tP-p-l,.p (r).

does not depend on

T.

From the expansion

up + 1

InO(u; p)=- p+ 1 it follows that for

up + 2 p+2 -

...

lui

(1.89)

pnO(u; P)I
r ~

IInJ(z)1 < 2r + p

1

p+ 1 ~

~

dn(t)

IP+l

F= _ _ 2_

p+ 1 rP

St,,+2 n(t) dt.

+1 n (v) +2r,,+1 (~r)P+l v

In other words GO

lin .. /(z)1 < 2Crp+l ftP(t)-,,-9dt. ow

64

GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

[CHAP. I

and by (1.53') [see property (3), § 12J co

r

1 ('Cr)' ("")-11- 1 p+ I-p

=

tP (t) - 11- 9 dt

•

+

0

«'Crt (~r)-Jl-1).

It follows that asymptotically

lIn f(z)1 ~

THEOREM

< p+l-p 3C 'Cp-p - 1rp(r).

25. If the sequence {at} of points on the positive axis has a density

!!Jr>

A= tim

(1.90)

r-+co rP(r)

for some proximate order p(r), if p is not an integer, and if co

II a(~; a/c p)

V(z) =

1c=1

then for z

= re i6 , 0 < 0 < 277, r

lim In V (re tfl ) = ~ elp (8-,,) -+ ex> r P (r) sIn 1tp •

and the limit is uniform in any interval 0

< 1J < 0 < 277 -

1J.

PROOF. It follows from Lemmas 7 and 8 that for arbitrary E > 0 and for sufficiently small (J > 0 and sufficiently large T > 0 we have asymptotically

~

l'n V(z)-

In

a (:/c; p)1 < ~ rp(r).

(1.91)

IW< ak <~r The sum on the left can be written as a Stieltjes

int~gral

~r

~

In

a (~;

I

p) =

","
In

0(;; p) dn (t),

qr

After integrating by parts we have

~

(:/c;

p)=n('Cr)lna(e:; p)

(or) In a (etfl;

p) - Zp+l I tP+l(t-z) n (t) dt

Ina

ar< a,,<""

~r

-

n

CI

ar

We use the asymptotic estimate n ('tr)

< (.1 + I!) 'CPrP

(r),

(6

=

arg z).

(1.92)

Sec. 17]

65

AUXILIARY FUNCTIONS

which follows from (1.90) and property (2) of proximate orders. From this estimate and inequality (1.89) it follows that for sufficiently large T we have the asymptotic inequality n('tr) lIn

a (':; p)1 < i rp (,,),

Similarly from the asymptotic estimate n (or)

< (A + e) oPrP (,.)

and from inequalities (1.21) and (1.87) we have the asymptotic inequality n(or) lIn

a (~; p)1 < i

rp (,,),

which is valid for all sufficiently small values of a. Returning to (1.92), we can now write

..,.

I

~

In a (~.

~

Qk'

p) + zlI+

1

~<~<..,.

It follows from (1.90) that for arbitrary ~ of t

In (t) -

and therefore for 0

J

1

I< ~rp(,,).

dt-AZlI+ 1

fir

MP L (t)1

t

~

(1.93)

< UpL (t)

f

dtl < 3rll+ 1 f

'tr

IP-P- 1L(/) I-Z

~

4

> 0 and for all sufficiently large values

'tr

n (t)

11'+1 (/-z)

For ar

n (/)dt tll+1(/_%)

< () < 277

'tr

,+1

f ~

t p- p - 1L (t)dt II-real

.".

Tr

(1.94) and therefore asymptotically rP+1

r

.

J "C'r

'If"

Ip-p- 1L

(t)dt

I 1 - re4' I

< 2rp+1L(r)

~

Ip-p-1dt = 2rpL(r)

I I - rea I

"

Thus, for 'YJ > 0 and 'YJ ~ the asymptotic inequality

(J ~

277 -

'YJ

AZP+1

f..,.

~,.

I f zp+l

n (I) dt tP + 1 (I _ relt)

Gf'

f ~P~:;:7 . "C'

II

and sufficiently small 15

t p- p - 1L (I) dt

t - rea

I<

> 0,

we obtain

(1.95)

..!..8 rP(").

Gf'

Using (1.94) once more we obtain

..,.

I f Azp+l

u

~

tp-p- 1L (I) dt - tlei(p+ 1)9 {

t-re i6

r

•

up- p - 1 du } r PL (r) u-Id

I

II

(1.96)

66

GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

Finally, for sufficiently small

(1

[CHAP. I

> 0 and sufficiently large 'T, 0:>

~

Ifup:::;u - fu;:~ dU/ < 8~' "

(1.97)

0

From inequalities (1.91), (1.93), (1.95), (1.96) and (1.97) it follows that asymptotically and uniformly in () for TJ ~ () .c;;;; 217 - 1]

fo ~:; dul < 0:>

/In V(re ifi )

-

tlrp(r)e i (p+l)I

er P(").

To complete the proof it only remains to compute this integral. 21 The reasoning we have used is not applicable in case p is an integer. In this case Lemmas 7 and 8 cannot be used to derive the asymptotic inequality (1.91). Also, (1.97) is not valid in this case since then the integral from zero to infinity does not exist. To study the case of integral p we prove the following lemma. LEMMA 9. Let the set {at} of points on the positive axis have density respect to the proximate order per) and let p be an integer (p ;;> I). Let

V,.(z)

a with

II a (~ ; p-1) ale>" IT a (:Ie ; p).

=

aJ:<"

Then for z = re i8, 0

< () < 217,

lim InV,.~e~) =-A[~-i(6-1t)]e'p8. p

r P( )

,,-'0:>

The limit is uniform in each interval 0 PROOF.

< 1] .c;;;; () .c;;;; 217 -

1].

We start from the equation

In V,. (z) =

~ In a(:Ie; P- 1) + ~ In a (:Ie ale > ,.

ak .:; ,.

It follows from Lemmas 7 and 8 that, for sufficiently small large 'T and all sufficiently large r,

(1

; p). > 0, sufficiently

,.

lIn V,. (z) -

f

In a (:

; P- 1) dn (t) ..,.

-J

In 0(-7;

p) dn (t) I

Q

gP-ldu

1

+,. .

Sec. 17]

67

AUXILIARY FUNCTIONS

or, after integrating by parts over 0

< () < 271, r

r

I

lnVr(Z)+.!..eiP'n(r)+zp net) dt P • t P(t- z) fit"

f

'tf"

+ zp+l

n (t) dt t P+ 1 (t-z)

r

1< .!...2 rP(r).

Just as in the proof of Theorem 25, we obtain from the properties of the function

1'(') = fL(t). for 0

< ij <; () <; 27T -

I" f

"ii. the inequality

r

zp

f 1

,,(t) dt -IlrP (r)eip'

t P(t-z)

or

t dtett -

I<

'8 rP (r)

"

and

I f zp+l

r 'O

'Or

n (t) dt

_ ArP(r)eiCp+l)'

tP+l(t-z)

< !..8 rP(r .

dt 1 . t(t-ett) 1

r

Also,

and 'O

1e i8 f t(t~ef8) + In (1- eil) 1< i· 1

It follows from these inequalities that for 0 asymptotically lIn Vr (re ie) + Ii [~ - i (6

< T/ <; () <; 271 -

T/ we have

-"It)] eiP·rP (r) I< erpr).

The function Vr(z) is not entire. It has discontinuities for Izl = at. However, we shall use Lemma 9 to construct an entire function of integral order with an asymptotic representation inside an angle analogous to the representation given by Theorem 25 . •1

Indeed 1

f t=IiiI = dt

1-e'8

In a _

eM =

In (I - e -II) -In (1 - ae-fIl).

" is valid for sufficiently small a. and the inequality

68

GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

[CHAP. I

THEOREM 26. Let the set {Ok} of points on the positive axis have density L1 with respect to some proximate order p(r), where p is an integer (p I). Let

>

co

U(l)

IT a (: ; p).

(z) =

"=1

-(~

00

u(lI)

(z)

= IT a (_:~l!_P_

"

"=1

"

and V(z)

= U(l) (z) u(~) (z).

Then, uniformly on each closed interval that is interior to one of the intervals 0< IJ < 'IT/p or 'IT/p < IJ < 2'IT, we have lim In ~ ~~~'8) rp ·

= !!:-. ~e(p'. p

r-+oo

Assume that p = p and represent the functions u(1)(z) and

PROOF.

in the form U(l)(Z)=ex p

(! zp a,,
and

u(~) (z) =

U(2)(Z)

exp ( -

! zP ~

ak P),

u~) (z).

ale
where

and U(2)

r

(z)

=

a (-z IT a,,
ale

II a (Z- --

("

e- P ;

•

P - 1)

(ft

a,,>'

ak

e

P;

We then obtain

v (z) =

U~I) (z) U~2) (z).

and to complete the proof it is sufficient to quote Lemma 9. For p = p + I it is necessary to represent the functions u(1)(z) and in the form Ul 1) (z)= exp

(-fzp

U(II)(Z)

~ akP)u~t)(Z) a,,>r

and U(2)

(z) = exp

(f zp

~

ai P)

U~2) (z)

Gl>r

and then to use the same lemma. In conclusion, we construct a function that will play the fundamental role in the study of the generalized indicator function.

Sec. 17]

69

AUXILIARY FUNCTIONS

LEMMA 10. of the angle

Let p(r) be a given proximate order (p

> 0) and let the opening

61 <: arg z <: 69 he less than the smaller of the numbers 7T/ p and 27T. Then for arbitrary real numbers a and b a function W(z) can be constructed that is holomorphic and has no zeros in this angle, and such that for 01 <; <; O2 we have uniformly

°

lim In I W (~~i1l) I = a cos p6 rP

+ h sin p6.

r -+00

PROOF. First consider the case of nonintegral order p. For arbitrary 11'1 and 1/12 satisfying the inequality 0 < 1/12 - 1/11 < 'TT / p, the system of equations

Al cosp NI +'It)+A 2 cosp ('~9+ 'It) = a, } A1 sin P (~1 + 'It) + A2 sin p (,~+ 'It) = h,

(1.98)

has the solution Ab A 2 , so that the determinant of the system sin p (~2 -1}1) =1= O. Choose 11'1 and 11'2 outside the interval 01 <;

Iii =

I

I

A, slnK'lt p and

lei

=

°<;

Oz. Let

sign (Ai sin 'ltp)

(i= I, 2).

Construct sets {ai ,l} and {ai,l} with respective densities .(\1 and .(\1' form the functions 00

V1 (z) =

II a (~; p), at, "=1

1

and, finally, form the function

W (z) = V:' (ze -i+,) V;- (ze -i<\l")

°

that is holomorphic in the angle 01 <; <; 0•. From Theorem 25 and equations (1.98) it follows that uniformly for 01 <;

°<; °z

lim In I W ~et'I) I = a cos p6 rP

r-+oo

J

+ h sin p6.

If p is an integer, then in place of (1.98) we consider the system

At c~s P';l Al

Sin

P'l1

+ A2 C~$ P';2 = + Ala PIjI, = Sin

b, }

a,

(1.98')

70

GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

where 0 < 1f'a - 1f'1 < 7T/ p and the four points 1f'1' 1f'1 outside the interval (JI ~ (J ~ Oa (for p ;;;. I and (Ja is possible). Then put (I

[CHAP. I

+ 7T/ p. 1f'a. 1f'a + 7T/ P lie (JI < 7T/ p such a choice

=

(1.99)

I, 2).

Construct sets {at .l } and {at.a} of points on the positive axis with densities ~l and ~a. respectively. and construct the corresponding functions VI(z) and Va(z) as in Theorem 26. From these functions we form the further function W(z} = V~' (ze-i<\l,)

V;' (ze-i<\l,),

holomorphic inside the angle (JI ~ arg z ~ (Ja. lt follows from Theorem 26 and equations (1.98') and (1.99) that uniformly for (JI ~ (J ~ 02 In I W (r,i~) I (1.100) lim (,., = a cos p6 b sin p6.

""00

+

rP

18. The Generalized Indicator The concept of the indicator of a function can be generalized by replacing the usual order by a proximate order. By a generalized indicator of the function j(z), holomorphic in the angle fit < arg z < fl2' we mean the function

where p(r) is some proximate order. Using the auxiliary functions of the preceding section we can obtain the following result. THEOREM 27. If p > 0, then the generalized indicator satisfies the fundamental relation h (6 1) sin p (6 i - 6.) h (6a> sin p (6. - 61) +h(6 s) sinp(61 -6 g) 0 (1.69')

+

<

< O. < O. and O. - 01 < 7Tlp. PROOF. Let 01 < 0 < 0•• 0. - 01 < 7T1 p. and let H.(fJ) = [h (6u + I) Sin p (6 - 81) + [h (68) + I) sin p (lis -

for 01

sin p (6s - 61)

6) (a

> O).

By Lemma 10 there exists a function W(z} having the proximate order p(r), holomorphic and without zeros in the angle 01 " arg z " 03 , and such that the function In IW (rei'!) I r P''''

Sec. 18]

71

THE GENERALIZED INDICATOR

tends uniformly to the function HiO) on the interval 0 1 oe;; 0 oe;; 0 3 , Thus the func· tion f(z) W-l (z) tends to zero along the rays arg z = 01 (j = 1, 3) and by the Phragmen.Lindelof theorem it is bounded inside the angle 0 oe;; arg z oe;; 0 3 , It fol· lows from this that, for 0 1 oe;; 0 oe;; 0 3 and for arbitrary e > 0, h(O) oe;; He(O). It reo mains to pass to the limit as e -+ O. It follows from the fundamental relation that the generalized indicator h(B) satisfies properties (a)-(h) of § 16 which were derived from the fundamental relation. Now we shall derive some general properties ofthe growth offunctions along rays, where the function has the proximate order p(r). The results will, of course, be true also in the special case p(r) == p, that is, for functions of normal type with respect to the usual order p.22a THEOREM 28. /ff(z) is holomorphic and has the proximate order per) (p > 0) inside the angle « <; arg z <; p, then to each ~ > 0 there corresponds a number r. such that the inequality

In I/(re i &) 1< (h (b)+ it) rP (f')

is validfor r

(1.101)

> r. for all values of 0 in the interval [«, P].

PROOF. Divide the interval [«, P] into subintervals with points of subdivision 01> Ba, ••• , O.._d« == 0o, P == 0..) and for each subinterval (0 1, 0HI) construct the trigonometric function HI(O) of the form (1.67), taking at the points BI and 0HI the respective values h(6j

>+;

and h(/jj+l)+ ; •

The intervals (OI ,OHI) can be chosen so small that the oscillation of the functions h(O) and HtO) on each of these intervals is less than e/3. Thus in each interval 2c HJ(fJ) h(b>+T' In each of these intervals we construct the function W tz) with indicator RI(O), whose existence was asserted in Lemma 10. The modulus of the function

<

I(z) Wjl(Z) is bounded on the sides of the angle (}I <; arg z <; (}HI' By the Ph ragmenLindelijf theorem it is also bounded inside this angle. Hence, for sufticiendy 22a If we

divide (I, 69') by p and formally take the limit as p ... 0, we obtain the inequality

,z>

h('.)('z - '3) + h('Z>('3 - '.) + h('3)('. <0 that defines ordUwy convexity. Griiin [I] showed that the generalized indicator of a function that is bolomorpbic and of order zero iDside an aqle II < , < fl is a convex fUDCtion of ,. Conversely, every convex function h('>' II < , < fl, is the indicator of a fUDCtion that is hOlomorphic and of order zero iDside the aqIe II < , < fl. SiDce the indicator of an entire ~ is ~ ~..-periodic fun~ of " Griiin'I rault implies GoI'dberz'1 rault [3] that the pneraIized indicator of an entiR function of order 0 is CODItanL

72

GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

large RJ(E) and r

> RAE),

Inl~(re")1 < [HJ{b)+ and if R(E)

==

[CHAP. I

max RAE), then for r

;]rp(r)

> R(E)

In 1I (re") 1

< (h(b)+a)rp(rl .

The following theorem on the connection between the type and the indicator follows at once from this theorem. THEOREM 29. The maximum value of the indicator h,(fJ) of the function f(z) on the interval at <; fJ <; Pis equal to the type al of this function inside the angle at <; arg z <; p.

Indeed, it follows from (1.101) that max h,(O) it is obvious from the definition of indicator that

•max

,,8,,~

h,(&)

> aI' and on the other hand

< a, .

THEOREM 30 (WIMAN). If the order p of an entire function is less than one, then there exists a sequence r1 < r2 < r3 < ... (r" - (0) such that for arbitrary E > 0 andn > n. »)0081 1M, (rn· where

PROOF.

Let co

I(z)

=Czr- II(l-~)

a"

11-1

be an entire function of order p

< 1.

Clearly

00

Cr·

00

IJ (I-I~I )
(Izl =r).

Let 00

11 (z) = Czra IJ (1- I:111 ) and let hl(fJ) be the indicator of the function!t(z). Since max hl(fJ) == h1(.".), it follows from property (f) of the indicator (§ 16) that if 0 h.(.".)cos 'lTp. Hence h.(O) > 0 if p < 1/2, and consequently we have. on a sequence r" too, the inequality 1011.(r,,)1 > 0 and also

Sec. 18]

72A

THE GENERALIZED INDICATOR

Hence in this case we obtain

-.- lnm(r) . InIJI(rn)1 Ii InIJI(rn)1 hl(O) hm ;> hm ;> m = -InM(r) 'ft- OO InIJI(-rn)1 'ft- OO hl(x)r:,<',> hl('IT)

:=

cos Xp.

HOO

Now let 1/2 " P < 1 and let IJ(zo>l = m(r). Then

In[ m(r)M(r)]

;> lnlj(zo)j( - zo)1 ;> lnl

IT (1 - lakl~

2

)1.

1

The substitution r2 = R returns us to the case of a function of order PI = P/2 < 1/2, and by the preceding discussion we find that

on some sequence rn too. Therefore

r

,~

lnm(r)M(r) lim lnIJI(r)jl( - r)1 InM(r) ;> '_00 hl('IT)"p<'> .

Hence, by the preceding discussion,

Again applying property (f) of the indicator, we obtain

l i lnm(r)

H~ lnM(r) ;> 2 cos

2

'lTP

T - 1 = cos 'lTp.

This completes the proof of the theorem.

Sec. 18]

73

THE GENERALIZED INDICATOR

The special case p < ! is especially interesting. In this case the theorem tells us that there is a sequence of circles Izl = rn on which m(r n) -+ 00, and the order of growth of m(r n) is the same as the order of growth of M(r n), while the type is ;> Of cos 'TTp.22b V. Bernstein proved the following precise theorem, giving an estimate of the density of the set of those r on which for fixed' fJ and e > 0 the inequality In I!(re'~) I> (h (6) - e) rP (r) is valid. THEOREM 31 (V. BERNSTEIN [2]). Let the function f(z) be holomorphic and of order p(r) in the angle a " arg z " p. To arbitrary positive numbers e > 0 and 0 < '" < 1, and each fixed ray arg z = fJ, there correspond a number 8 > 0 and a sequence of intervals rn " r " rn(I + 8) (rn -+ 00), on each of which the inequality In If(re i9 ) I [h (6) -a) rP (r) is satisfied except perhaps on a set of measure not exceeding w~rft'

>

PROOF.

To each y

Without loss of generality we may asslime that fJ = 0 and h(O) = > 0 there is a sequence rn -+ 00 such that

In I f(r n) I> - yr"j-r.). Also, if we choose ~ > 0 sufficiently small, then for r > ry,'" depending only on ~ and y,

o.

IfJl < arc sin (2e~) and

In If(rei~) 1< (h (6)+ "() rP (r) < 2y,.,,n Z f(rn) •

Then CPn(O) = 1 and for Izl '" 2e8rn In I ~n (z) 1-< 3"( (r Iz I)p(rn + I -' I. It follows from Theorem 11 that the inequality

n+

In I ~n (z) I > - 3"(H

(=) (rn + 2eern)P

(rn + 2ewnl

is satisfied in Izl '" ~r n but outside exceptional circles the sum of whose radii is not greater than w~r n' Returning to the function f(z) we see that the asymptotic inequality

lnlj(r)l>

-y[I + 3H(-i-)(l +2eO)2p]r~(rfll

is satisfied on the whole interval (1 - 6)r. < r < (1 + 6)r., except perhaps for intervals the sum of whose lengths is less than 2w6r.. Using properties of the 22b There are many papers about Wiman's theorem. For a generalization of Wiman's theorem to meromo~c functions see GoI'dberg and Ostrovskii [1).

l2c The continuity of the indicator when p - 0 follows from a theorem of Griiin (see footnote at the end of 16 in Appendix VIII).

74

GENERAL THEORY OF THE GROWTIf OF ENTIRE FUN('''TIONS

function rP
[CHAP. I

< r < (1 + 6)r. we have

> (1 +o)-2PrPlrl.

To complete the proof it is sufficient to choose y and b so small that

r[ 1 + 3H( ~)(1 + 2e8)2p] < e(l + 8)2p. 19. Plane Convex Sets

For p = I the indicator h«() has a simple geometric interpretation. To present it we shall need the basic properties of plane convex sets. A convex set is a nonempty closed set which contains the entire line segment joining each pair of points in the set. In particular, a line segment is a convex set. A single point is also considered to be a convex set. Clearly a nonempty intersection of convex sets is again convex. The intersection of all the convex sets containing a given set is thus the smallest convex set containing the given set. It is sometimes called the convex hull of the given set. It is easy to see that the convex hull is also equal to the intersection of all the half-planes containing the given set. We introduce the operation of arithmetic sum of convex sets. By the sum G1 + G2 of two convex sets is meant the set of all points of the form z = Zl + Z2' where Zl E G1 and Z2 E G2. Let us show that G1 + G2 is a convex set. Let z~ + z~ and z; + z; be two points of this sum. Since

+z;) = AZ~ +IlZ; + (A + II = 1, A>- 0, !l >- 0),

Z = A(~+z;)+!l(z:

AZ;+!lZ;

and since ).z~ + !-,z; E G1 and ).z~ + !-,z; E G'I. by the convexity of G1 and GI , we have Z E G1 + G'/.. It is clear that G1 + G2 is closed. We shall always assume in this section, without explicitly mentioning it each time, that our convex sets are bounded. An important concept is that of the supporting function of a convex set. By the supporting function of a convex set G we mean the function k(fJ)= sup (xcos6+ysin6)=supRe(ze-"). z+illEO

(1.102)

_EO

Since G is closed and bounded, this supremum is attained at some point of the convex set G. The line

x cos 6

+ y sin 6 -

(1.103)

k (6) = 0

will be denoted by 18, Clearly the line 18 has a point in common with G. Further, all points of G lie on one side of each such line, since from the definition of support function k(O) we have, for x + iy E G, x cos (j y sin (j - k (fJ) <: O.

+

Sec. 19J

75

PLANE CONVEX SETS

The lines (/0) are called the supporting lines of the convex figure. A line of support is clearly a generalization of a tangent. The points that a support line has in common with the convex set are called the supporting points of the given line of support. A line of support has either a single point of support. or a whole line segment of supporting points. Those boundary points of G that are not interior to any straight line segment in the boundary are called extreme points of the convex set. It is easy to see the geometric meaning of the supporting function. It is equal to the distance from the supporting line to the origin. Clearly the supporting function of a single point Zo = P exp (i0o) has the form k(O) = p cos (0 - ( 0 ), The supporting function of the segment -H <; Y .;; H on the imaginary axis is k(O) = H Isin 01. Denoting by k(fJ) the supporting function of the sum Gl + G2 + ... + Gft of convex sets. we easily verify that

k(6) = k t (6)+k 2 (6)+ . " +kn(f}).

(1.104)

where k;(O) (j= 1.2•...• n) are the supporting functions of the convex sets Gt> G2• •••• Gft' For the intersection of convex sets we have 23 (1.105) For the convex hull of the union of the convex sets Gl • G2• supporting function is given by

••••

Gft the

(1.106) Let us prove this last assertion. Any half-plane that contains all the sets G1

(j = 1,2•... ,n) is given by an inequality xcos6+ ysin6 -k(fJ)

< 0,

where k(O) ;> max {kt(O), k.(O) • ...• kft(O)}. Since the convex hull is the intersection of all such half-planes, we obtain (1.l06). The following theorem gives a characterization of supporting functions of convex domains. THEOREM 32 .. A necessary and sufficient condition thaI the function k(O) be the supporting function of a bounded cont'ex set is that the following conditions be fulfilled:

"' Translator's nou: this should be an inequality, k(fI} '" inf {k,(fI}}. For example. consider the intersection of the two convex sets -1 '" Y '" 1 and -1 '" x '" 1. The supporting functions are Isin 91 and lcos 91. whereas the supporting function for the intersection is O.

76

GENERAL THEORY Of THE GROWTH Of ENTIRE fUNCTIONS

(a) k(fJ

(b)

+ 2,,) =

k(fJ)

(- 00

[CHAP. I

< fJ < (0),

k (8 t ) cos 6J sin 61 k (611) cos 82 sin 82 k (6 3 ) cos 03 sin 83

>0

" 93, 93 - 92 < '/T12, 92 - 9. < '/T12. The expanded form of condition (b) can be written as follows:

for all 9. " 92

k (81) sin (/):! PROOf.

z= x

8s)

+k

(/}2) sin

(6s - ( 1)

+ k (0

3)

sin (/}1 - ( 11)

< o.

Let G be a convex set, let k(O) be its supporting function, and let

+ ;y be a point of the boundary of G belonging to the supporting line

I•. Then we have three relations:

•

+ y sin 6 k (6 < o. xcos8 + ysin811-k(8~= 0, xcos6 s + ysin6 a -k(6 s) < O. x cos 6J

J-

J)

11

Multiplying the first of these by

(1.107) the second by I

cos 63 sin 63 1

.

COS 81 SID 61

= SID. ('/1 f1

6

3)'

(1.108)

>0

(1.109)

the third by

I

I

cos 81 sin 81 = sin(811 -81 ) coc611 sin 611

and adding, we obtain (b). Condition (a) is obvious. Let us prove the sufficiency. We choose 0. arbitrarily and construct the line I•• : xcos 81

+ Y sin61-k(6~= O.

Without loss of generality we can assume that 0. The open half-planes

(1.110)

= ,,/2. (1.111)

meet 1"/2 in the intervals (bl ,

+ (0), whose union is the interval (b, + (0).

Sec. 19]

77

PLANE CONVEX SETS

Similarly. the union of the intersections of the half-planes (1.112) with the line 1,,{2 is the interval ( - 00, a). We now show that no point of the line 1,,{2 can belong to both of the intervals ( - 00, a) and (b, + (0). Indeed, if there were such a point then its coordinates would satisfy (1.110), (1.111) and (1.112), and multiplying them, respectively, by (1.108), (1.107) and (1.109) and adding we would obtain a relation contradicting (b). Consequently, a <; b, and all points of the interval a <; x <; b on the line I"{I belong to all the half-planes xcos6+ y sin fj -

k(6)

<: O.

Thus the intersection of these half-planes is not empty, and therefore it is some convex set G. This set lies on one side of each of the lines x cos 0 + y sin 0 k(O) = 0, and has points of contact with each of these lines. Thus these lines are supporting lines for the convex set G, and k(O) is the supporting function of this set. It follows from Theorems 23 and 32 that the indicator function of an arbitrary entire function of the first order (with respect to some proximate order per) - I) is the supporting function of some bounded convex set. This convex set is called the indicator diagram of the given function. An arc will be called convex if it lies entirely in the boundary of its convex hull, and locally convex if each of its points has a convex neighborhood on the arc. From the preceding considerations it is clear that condition (b) is necessary and sufficient for the convexity of an arc. A locally convex arc can intersect itself. Clearly the indicator function of a function that is analytic and of the first order inside an angle of opening less than or equal to .". is the support function of some convex arc. In the general case, if h,(O) is the indicator function of some function holomorphic and of order p in some angle, then by (1.69) the function h,( 0/ p) is the supporting function of some locally convex arc. Iff(z) is an entire function of integral order p, then the corresponding locally convex curve is closed, and in traversing this curve the supporting line makes p complete rotations. Let us study convex sets in more detail. We first show that through each boundary point there is a supporting line. Since the assertion is obvious for sets consisting of a single point or of a line segment. we limit ourselves to sets having interior points. If the extension of some ray through the boundary point Zo of the convex set G passes through the interior point Zl' then clearly there is some angle at Zoo containing this ray. and containing no points of G (Fig. 4). A boundary ray of this set ofrays and its extension form a line I that can contain no interior

78

GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

[CHAP. I

Fig. 4

points of G. The entire set G lies on one side of I, since if there were interior points of G on one side of the line I and if there were also points of G on the other side, then there would be interior points of G on /. Thus / is a supporting line to G through =0' The boundary r of a convex set has a finite length. This follows from the fact that the perimeter of any polygon (inscribed in the boundary of the set and having no self-intersections) does not exceed the perimeter of any triangle formed by three supporting lines. If the origin is inside the domain and if arc length on the boundary is measured counterclockwise from some fixed point to the point of support, then the angle 0 is a nondecreasing function of arc length s. Indeed. let ZI and Z2 be two points on the boundary of G, with arg Z2 > arg ZI' The line segment joining ZI and Z2 belongs to G and therefore lies on the same side of the supporting lines at the points ZI and Z2 as does the origin. It follows that the supporting line at Z2 has been rotated through a non-negative angle with respect to the supporting line through ZI. that is, O2 ;> 01 , Clearly at angular points of the boundary the function 0 = O(s) has jumps, while on line segments in the boundary O(s) is constant. The inverse function s(O) has jumps when the ray arg z = 0 is perpendicular to some line segment in the boundary and is continuous for all other values of O. The geometric interpretation of trigonometrically convex functions enables us to obtain the analytic properties of these functions that were established earlier. We shall obtain geometric interpretations of the quantities 8.

k~(6). k~(6) and s(61• 69)=k~(6:J-k~(61)+ fk(a)d6 8,

of which we spoke in § 16. Let us show that k(O) is a continuous function. Without loss of generality we can assume that the origin lies inside the convex set G. We denote by tS(~O) the length of the segment of the ray arg z = 0 + ~O between the supporting lines /" and 16+116 (Fig. 5). We then have k (0

+ AD) = k (6) sec (AO) + a(AO).

(1.113)

Sec. 19]

PLANE CONVEX SETS

79

To establish continuity it is sufficient to observe that 13(110) 1< d tg(M), where d is the diameter of the domain. We now investigate the differentiability of k(O) and we find the geometric significance of k'(O). Let p(O) denote the segment of the line Ie froni the point of support to ttoe foot of the perpendicular from the origin to the line Ie. It is easy to see that the radius-vector to a point of the curve r is a continuous function of sand consequently, at the extreme points, it is a continuous function of 0. It follows that p(O) is continuous at the extreme points. If the direction 0 is perpendicular to some segment on the boundary then we have

r.

p (0

+ 0) -

p (0 -

0) = d (0).

where d( 0) is the length of this segment. From simple geometric considerations (see Fig. 5) we have the inequality tg (M)(p(fJ+AfJ)-Asl<:3(AO)

<.p (r+AO)tg(AO).

From (1.113) and this inequality we see that at each extreme point k' (0) = P (0).

At points of discontinuity one establishes in the same way the existence of left and right derivatives of k(O), with k~ (0) = p(O+ 0),

k ~ (f.I) = P (0 - 0) .

We now find an expression for the length s of the arc between the points of support of the lines 1m and IfJ in terms of the supporting function k(O). We inscribe in this arc a polygonal line :E with such small segments that

Ik(fJ)- R(fJ) 1< 8, where R(O) is the support function of the inscribed polygon :E.

Fig. S

80

GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

[CHAP. I

Letting dt denote the lengths of the segments, we have

"

"

11 .. 1

11-0

s, = ~ dIe = ~ IR~(6,,)-R~ (6,,»)

Here Ot are the directions perpendicular to the segments of the polygonal line. Note that in the intervals (Ot, 01<+1) the function R(O) has the form At cos (0 - 0t); we have n-l

5t

= R~ (6,,)- R~ (1J 0 ) - ~

k=1

or 8n

5t

= R~ (6n)-R~ (6 0

)+ f R('J)d6 90

and, passing to the limit as max dt

--+

0, we have

50-+-0)-5(1% -1-0)= k~ O)-k~(I%)+

~

f.

k(6)dO.

(1.114)

Note that for the length of the entire boundary of the convex set we obtain the formula ll6 21<

5

=

f

(1.115)

k(lJ)d6.

o

From the monotonicity of s(0) it follows that the radius of curvature R = tis/dO exists at all points 0, except perhaps for a set of measure zero, and that R

=

k" (6)+ k (6).

Comparing formula (1.114) with (1.77) of § 16 we see that if p = I the function $(0) defined by (1.77) is the length on the boundary of the indicator diagram of the function from some fixed point to the point of support of the line I". If p ~ I and if a(0) denotes the length of the locally convex arc with support function k(O/ p), then from (1.114) we have the following equation:

a(p~-+-O)-a(p(l-+-O)=+{k;O)-k;«I)+pll

J k(6)d~} ~

.

.. If the curve r has constant width, that is, if k(8) + k(8 + ..) = d(O" 8 (l.lIS) we have s - mi. This is a well-known theorem of Barbier.

•

< 2..). then from

Sec. 19]

81

PLANE CONVEX SETS

Comparing this with (1.77) we see that for p

>0

s (6) = po (p'1). We now solve the converse problem of expressing the support function k(O) in terms of the arc length s(O). Put the origin at the point of support of some supporting line 10 and inscribe a polygonal line between this point of support and the point of support of the line Ie. Clearly the support function of this polygonal line is n

R (6) =

~ die sin (0 -

Ole).

k-I

where 0" are the directions perpendicular to the segments in the polygonal line, d" are the lengths of the segments and On <;; 0 < 8n+l' Passing to the limit as max d" -+ 0, we have 8

k (6) =

f sir. (6 -

4) ds (It).

o Translating the origin to an arbitrary point in the plane and using the formula for the support function of the sum of two convex sets we have

k(6)=Acos(6-6o)+

f•sin(6-i')ds(4)·

(1.116)

o Formula (1.116) enables us to associate a convex set to each monotone function s(tp). The closure condition k(O + 217') =-k(8) can be written in the form 2K

f sin(0-I\I)ds(4) = O.

o

2K

f e·"'ds(4)=O.

o

Similarly one can construct a locally convex curve L with support function k(8/p) for a given monotone function s(6)=k'(fJ)+pll

f•k(cp)dcp. o

IS In Theorem 24 a different representation of the function k(6) than (1.116) was given for a closed convex curve

1(8) = A cos (8 -80)

+

f•(8-;) sIn (8

1-211

-4)ds (;).

82

GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

[CHAP. I

In this case formula (1.116) is rewritten in the form

•

)+ Jsjnp(6-~)ds(I~).

k(8)=Acosp(8-fJo

o and the condition of closure of the curve for integral p takes the form 21<

f

eip'"

dS(I~) =

o.

(1.117)

o

Using this geometric interpretation of condition (1.117) we can solve the following problem. Let a nondecreasing function 5('1') be given such that for some integer p z,.

f

eiprjo ds (I~)

= o.

o

Find a nondecreasing, piecewise constant function sl'p), approximating s(tp) to within E > 0, that is, (O~ 1\1

< 21t).

We require further that the jumps of the function s.,,(tp) be rational numbers, not exceeding a given positive number IS and such that the function s.,,(tp) satisfies condition (1.117). To solve this problem we construct the closed, locally convex curve L corresponding to the given nondecreasing function s(tp), and inscribe a closed polygonal path L." in it in such a way that the segments in L." have lengths not exceeding 8, the length of L, differs from the length of L by less than E, and the length of each piece of the path whose endpoints are at points of L, is less than E. By shifting the vertices of L, slightly we can require in addition that the lengths of the segments be rational. The function sil/l) corresponding to L, gives the solution to the problem. In Chapter II we shall use this solution. To conclude this section we note some almost obvious properties of the indicator diagram of entire functions of finite order. 1. If the entirefunctionf(z) is of order one and normal type (that is, p(r) - 1), then the indicator tliDgram of the function tp(z) -

es a fez)

(a is a complex number)

is obtained from the indicator diagram I, of the function f(z) by a translation through the vector 0. Indeed, clearly

Ia Icos (6 + ex) + A,(fJ) (ex = arg a). The function 1411 cos (8 + «) is the support function for the convex set consisting h,(8) =

of the single point I.

Sec. 19]

PLANE CONVEX SETS

83

Thus h,lf) is the support function of the arithmetic sum of the convex sets If and ii, or

1,=1,+0,. 2. The indicator diagram of the product of two functions is contained in the arithmetic sum of their indicator diagrams. 26 This follows at once from the fact that the indicator of the product of two functions does not exceed the sum of their indicators. 3. The indicator diagram of the sum of several functions is contained within the smallest convex set G containing the indicator diagrams of all the summands. This follows from the inequality h(,+f,+" .+'n (6)

-< max [hr, (6).

hr. (Ij), ... , h'n (6)].

We now sharpen this theorem somewhat. Let E denote the set of those extreme points of G that belong to just one of the indicator diagrams. Clearly, each isolated point of this set is the common end point of two straight line segments in the boundary of G. The smallest convex set containing E will be denoted by I(E). 4. The indicator diagram of the sum of several functions contains the set I(E). This result follows easily from the following fact noted in § 15; if for some value of () the indicator of one of the summands is larger than the indicators of all the others, then for this value of () the indicator of the sum function is equal to this largest indicator. Now suppose that Zo E E is not the end point of some line segment in the boundary of I(E), and let ()o be a direction for which Zo is a point of support for the line 160' By the definition of the set E the indicator of one of the summands is larger than all the others for () = ()o, and hence Zo is in the indicator diagram of the sum. If Zo is the common end point of two straight line segments in the boundary of I(E), then from among all the directions () corresponding to the angle point Zo we can choose one direction ()o for which the indicator of one of the summands exceeds the indicators of the others, and hence Zo is contained in the indicator diagram of the sum. Finally, points of E that are end points of just a single straight line segment in the boundary of I(E) are limit points of other points of E. Hence the closed convex hull of the points of E of the first two types contains these points, and thus coincides with I(E). Thus the indicator diagram of a sum of functions contains I(E). In particular, if G and I(E) coincide, then the indicator diagram of the sum is equal to G, namely, the smallest convex set containing the indicator diagrams of all the summands. Such will be the' case, for example, if the indicator diagrams of the summands are obtained from one another by means of nonzero •• Further on (Chapter III) we shall prove a much more precise theorem in a special case.

84

GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

[CHAP. I

translations. In particular, if we multiply the function fez) of order one and normal type by a finite sum of the form n

'P (z) = ~ aki~ , kal

where Ok and Ak (k = I, 2, ... , n) are complex numbers, distinct, then the indicator diagram of the product

Ok

F: 0, and all

Ak

are

'P(z)/(z)

is the arithmetic sum of the indicator diagram of fez) and the smallest convex l". polygon containing the points AI' A2 ,

la, ... ,

20. Entire Functions of Exponential Type Entire functions of at most the first order and normal type are called entire functions of exponential type,27 and the exponential type of the function is the quantity -.- InM,(r)

a=hm

r

.

Thus functions of order less than one, or of order one and minimal type, are said to be of exponential type zero. Functions of exponential type occur frequently in the applications, especially in harmonic analysis and in boundary value problems in differential equations. To each entire fUllction of exponential type co

/(z) = ~ an z" ~ nl

( 1.118)

,,=0 there corresponds a function co

'P (z)

= ~ ~~l'

( 1.119)

"=0 called the Borel transform of the functionf(z). If (1 is the exponential type of the function fez), then by formula (1.06)

a=lim

Vla,,1

and thus the series (1.l19) represents a function holomorphic in the domain Izl > (1. The smallest convex domain I, containing all the singularities of !p(z) is called the conjugate diagram of the function fez). It clearly lies inside the circle Izl <; (1. P61ya [IJ established the following remarkable theorem . .. Translator's nou: In the Russian literature such functions are said to be of "finite degree."

Sec. 20]

ENTIRE FUNCTIONS OF EXPONENTIAL TYPE

8S

THEOREM 33. The conjugate diagram of an entire function of exponential type is the reflection in the real axis of its indicator diagram.

PROOF.

The proof depends on two integral formulas. The first represents

fez) in terms of its Borel transform !(z) =

~,

J

etl
(~) dC.

(1.120)

where C is any contour containing the conjugate diagram. Indeed, the integral on the right is unchanged if we replace the contour C by any contour containing the disc Izl <; (1. But for such a contour !F(z) can be represented by the series (1.119) and integrating termwise we obtain (1.120). The second formula gives an integral representation for !F(z), co

= f e- 8t!(C)dC.

(1.121)

o

where the integration is along the ray' = re-iIJ, t > O. It is easy to verify that this integral converges absolutely and uniformly in the domain (1.122) Reze- i8 h,(-6)+a.

>

The convergence of the integral follows at once from the asymptotic inequalities

Ie-de- i8 I< e -lla,(-81+e1 e and [ Ia

I!(tr ia ) I < e '

{-81+.!.]

e

2.

To prove (1.121) it is sufficient to show that the equation is valid in a part of the domain (1.122), for example, for Re Ze- il

> 30.

We show that for these values of z the series (1.118) can simply be substituted into (1. 121) and integrated termwise. Indeed, in that half-plane

Irife-iII < ra., . On the other hand, from the inequality

la,,1

M,{r)

1iI
R,,(t)

= ~ : t-II+l

tt

(1.123)

86

GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

[CHAP. I

of the series (1.118) satisfies the ineq uality IRn(t)1

Mf (r)

(t )11+1 '

< -1--t- r r

and if we put r

= 2/, then

I Rn(t) I < (;)" e2 (a+o)t.

(1.124)

From inequalities (1.123) and (1.124) it follows that the series co

" -.t a"C· ~e Ii! k=O

converges uniformly on the whole ray 1 > 0, , = te- iO • Integrating term wise we obtain equation (1.121). We now use formulas (1.120) and (1.121) to prove the theorem. Let kf(O) be the support function of the conjugate diagram If. In formula (1.120) choose the contour of integration so that it is no further than s6me E > 0 from the conjugate diagram. Then we obtain the estimate

If(re i9 ) I

,. max [Re (te i9 )J

< Mle

~EC

[k (-9I+oJ" '" Mle f

(M. =

2: tE~ 1

!{I (C)

I)

where c is the length of C. Hence

h,(6) ~ k,(- 6). On the other hand, from the convergence of the integral (1.121) for Re(ze- i9 )

> h,{-6)+_

it follows that 11'(=) has no singularities in this domain, and hence k,{fJ) '" h,{- 6). Thus

k,{- fJ) = h,(6). and the theorem is proved. We present some examples. If 11

f(z)

then

= ~ a"el"f.

"-1

Sec. 20]

ENTIRE FUNCTIONS OF EXPONENTIAL TYPE

87

and the conjugate diagram is the smallest convex set (polygon) containing all the points )'1:' If

"

f(z)=~el.~(akO+aklZ+'" +akPlezPIe}. then

k-l

and the conjugate diagram is the same as in the first example. We construct an entire function of exponential type having for indicator diagram a convex set I given in advance. 28 Let 1 denote the set symmetric to I with respect to the real axis. Choose a countable set of points AI:' dense in the boundary of I, and form the function CIO

!p (z)

:=I

ale ~ Z-II.Ie

--"

k .. l

with

This function is holomorphic outside the set I. We now show that all the points AI: are singular points of this function. Indeed, let z approach the point Ap along the perpendicular to one of the lines of support through the point Ap; then, clearly,

IZ-Apl < IZ-Akl

for

p =1= k.

Choose the number N p so that

~ I ak I < ~ I ap I· k>Np

We have

or I a,,1

I , (z) I > 2 Iz _ >.." I •• In Chapter I I we solve the corresponding problem for functions having an arbitrary proximate order.

88

GENERAL THEORY OF THE GROWTH OF ENTIRE FUNCTIONS

[CHAP. I

Clearly as =-+ Ap in the manner described, Iq;{z) I -+ 00. Thus I is the smallest convex !tet containing all the singularities of the Borel transform 9?(z) of the entire function 00

f(z)

= k=l ~ a,/"k·.

Note also that Polya's theorem enables us to sharpen assertion (4) about the indicator diagram of the sum of entire functions, provided that the functions are of exponential type. Namely, we can assert that the indicator diagram of a sum of entire functions of exponential type contains the smallest convex set containing all those extreme points of each of the indicator diagrams that do not lie in any of the other diagrams. Jndeed. each extreme point of one of the conjugate diagrams is a singularity of the corresponding Borel transform. If such a point does not lie in any of the other diagrams then it is a singularity for the sum of the transforms, and hence it is in the conjugate diagram of the sum of the given entire functions of exponential type.

CHAPTER

II

ENTIRE FUNCTIONS WHOSE SETS OF ZEROS HAVE AN ANGULAR DENSITY 1. Basic Results In this chapter we shall consider a special class of entire functions whose zero distributions have a certain type of regularity. This class is met quite frequently in various applications and plays an important role in the following chapters of this book. We shall give a precise asymptotic representation of the functions of this class. The regularity of the distribution of the zeros will be characterized· by a certain type of density of the set of zeros. 1 The proofs of the majority of the theorems of this chapter are somewhat involved. In order that the reader may not lose the perspective of this chapter as a whole, we shall present all the basic results of this chapter in the first section without proofs. During the first reading of this book the reader may confine himself to reading the first section only; he should, however, return later to the other sections that contain the proofs. We shall say that the point set 91 of the complex plane has an angular density of index p(r) if for all but a denumerable set of values f} and 8 (0 < f) < 0 ~ 21T) there exists the limit 2 (A)

A(CI 6) =

u

'"

\.

1m

r~oo

n (r, &, 6) rP

(r)· ,

(2.00)

where n(r, D, 0) is the number of points of the set 91 within the sector Izl <; r, f} < arg z < O. The quantity /:l.(D, 0) will be called the angular density of the set 91 in the angle D < arg z < 0, or, more simply, in the angle (f), 0). For a fixed D, the equation tJ.(6)-tJ.(fi) =tJ.(n, 6)

(2.01)

determines, to within an additive constant, a nondecreasing function /:l.(8). This function is defined for all values of 0 except, possibly, for a denumerable set of values. The exceptional values of D and 0 for which there does not exist 1 The principal results of this chapter have been published earlier by the author [1; 2). Analogous results were obtained independently by A. Pfluger [1; 2; 4). • When we say that a limit exists, we shall always mean a finite limit. 89

90

ENTIRE FUNCTIONS WITH ANGULAR DENSITY

[CHAP. II

an angular density a( 1'} + 0, 0 - 0) must be points of discontinuity of the function Indeed, if 1'} and 0(0 > t'}) are two arbitrary numbers, and if the positive number e is such that t'} ± e and 0 ± e are points of continuity of the function then

ao.

ao,

tJ.. (fJ _ a) _ tJ.. (I)

+ e) ~

lim n (r

'" r

-+ OJ

t

~ lim n (r

at 8)

rP (r)

"" r

-+ co

t

at 6)

rP (r)

<- tJ..('J +a) -

/},(fI- z).

ao,

If ~ and 0 are points of continuity of then the lower limit is equal to the upper limit. A set will be said to be regularly distributed relative to p(r) if it has an angular density p(r) with p not integral. We shall show in this chapter that the regularity of the distribution of the zeros of an entire function implies a certain regularity of the growth of the function. In order that we may be able to describe precisely the nature of this regularity of growth, we introduce now still another concept. The linear density of a set C of circles (C;) in the complex plane is defined as the limit

p. (C)

=

lim

r-+co

{..!.. '\'I r r ~

ril,

where r; is the radius of the circle C;, and the symbol L r indicates summation over all circles whose centers lie in the circle Izl < r. If the limit p*( C) does not exist then we shall consider the upper and lower limits of the same quantity, and we will denote these upper and lower limits by P*(C) and ~*(C), respectively. We shall call them the upper and lower linear densities of the set of circles C. A set whose linear density is zero will be called a CO-set. The asymptotic equalities which we shall establish are related to the order of growth. By the asymptotic equation f(r)

~ r.p

(r)

we shall mean the fulfillment of the following condition: lim

r-p(r)

[/(r)-!p(r))=O.

THEOREM I. Let p(r) be a proximate order of growth, let p = lim p(r) be a non integral number, and let {an} be a point set of the complex plane whose angular density ~('P) has the index p(r). Then for every z not belonging to some CO-set, the entire functioll co

f(z) =

z"'eP(a)

II a (a:; p) 1

(P(Z)=CO+C1Z+ ... +CpZ".

(2.02)

Sec. I]

91

BASIC RESULTS

will satisfy the asymptotic equation In

I

If(rei~) ~

H(O)rP(rl,

(2.03)

in which 8

r

H(fJ) = 'ItI s n'ltp ,

cos p (6

-~ - 1t)d!l('~).

(2.04)

8-21<

Here the function h" r (0) =,-P (r) In If(re ia ) I

will tend to the limit H(O) uniformly relatire to 0 when 0 '" 0 '" 21T. Hence, for a function of a nonintegral order, the existence of an angular density for the set of zeros implies a special typc of rcgularity of growth which is expressed by the uniform convergence to a limit of the function h,.,(O); the function H(O) does not depend on the factor in front of the canonical product, and is therefore completely determined by the regularly distributed set of zeros. For functions of integral order, the existence of an angular density of the set of zeros is, however, not a sufficient condition for such a reqlllarity of the growth of the function. In this case there is also required a special type of symmetry in the distribution of the zeros of the function which is characterized by the existence of the limit (B)

. 8, (r)

8,= ro+ex> hm -L() r

+!

(a,(r) = cp

~ a;p), 10" I
where L(r) = rP(,)-p, and c p is the coefficient of the term of highest degree in the exponential factor in front of the canonical product in the representation (1.38). We shall refer to this condition as condition (B). If P = p = per), then this condition indicates a special conditional convergence of the series

namely, the existence of the principal value

of the sum of this series. For functions of integral order we shall prove the following theorem. THEOREM 2. Let p(r) be an order of growth, let p = lim,..... "" Per) be a positive integer, and assume that the set {ak } of the zeros of the entire junction ex>

f (z) =

z"'eP ,.)

IT a (,,:; p) 1

has an angular density with index p(r), and satisfies the condition (B).

92

[CHAP. II

ENTIRE FUNCTIONS WITH ANGULAR DENSITY

Then for al/ z not belonging to a certain CO-set, the following asymptotic equation will hold

In I!(re ie ) I ::=:: H (6) rP (r),

(2.05)

where

H(fJ) =

-1

9

(,?-O)sinp(,?-f.l)dA{t})+'t.tCOip(6-6,)

9-2x

(2.06)

.

and tJ f = 'Tf e i8 ,. Here the function

h" r(6) = r-P (r) In I!(re lt) I

will tend to the limit H«() uniformly relative to () when 0

<: () <: 21T. 3

In view of this theorem, it is natural to define a regularly distributed set with index per), when p is not an integer, as a planar point set which has an angular density for this per) and which is such that for some constant Cm the following limit exists: (D')

a= lim

L ~r)

{e!/!+ ~ ~

r -+ 00

a;p}

I an I " r

(an E 9l,

L (r)

=

rp(r)-p).

If L(r) -+ 00, or has a limit different from zero when r -+ 00, then in the condition (D') the quantity C~l may be replaced by an arbitrary number. In the opposite case, the number C!II is uniquely determined. From the entire functions whose zero sets coincide with 91 we select the canonical functions f(z; 91). If p is not an integer we shall say that a function is a canonical function if it has the form 00

fez; 9l)= ~ 0

(a:; p)

(p

=

[p)),

while if p is an integer, a function is said to be a canonical function if it is of the form 00

!(z; 9l) =

e{J!JICP

~0

(a:; p).

Here, if the quantity C91 in condition (D') is arbitrary, we take Cm = O. The quantity H«(), defined in the statements of Theorems I and 2, is completely determined by the set 91 if p is not an integer, or if p is an integer but the quantity em is uniquely determined. In general, we shall denote the function H«() for canonical functions by H m«(), and we shall call it the indicator of the set 91. A function j(z) whose roots are regularly distributed can be evaluated within lB. Ja. Levin [1, pp. 1121-1127], A. Pfluger [41.

Sec. I]

93

8ASIC RESULTS

the exceptional circles by the principle of maximum modulus, and one obtains for the indicator h,(O) of an entire function the equation h,(fJ) = H (~).

The formulas (2.04) and (2.06) thus yield exp~essions for the indicators of entire functions with regularly distributed zeros in terms of the angular density and of the quantity lJ,. In Chapter l,§ 16, we derived similar forms for indicators; but it is important to note that in formulas (2.04) and (2.06) in place of the nondecreasing function s,(lJ') we have A,(lJ'}, the density of the set of zeros of the entire function/(z) in the angle (0, 11'). From the inversion formulas of § 16, Chapter 1, it follows that &,('~±O)

1 = 27tp s,N±O)+C

(C is a constant) (2.07)

or

1l,(1j -+- O)-&,(f) -+-0) =

~p

[s,(1j ± 0)-s,(3 -+- 0»).

Starting from the geometric interpretation of the function s,(O) (see § .19, Chapter I), we find that if p = 1, the angular density of the set of zeros of the function/(z) in the angle (D, 0) is equal to t1T times the length of the arc of the indicator diagram between the points of support of the supporting lines I. and III' which are perpendicular to the sides of the angle. In the general case, the right hand side of (2.07) is proportional to the length of some locally convex arc (Fig. 6). The formulas (2.07), (2.04) and (2.06) permit one to construct an entire function of any given proximate order p{r) in such a manner that its indicator will coincide with an arbitrary trigonometrically convex function h(O) of period 21T. The essence of the method is the construction of a nondecreasing function A(8) for the given function h(O) with the aid of equations (2.07) and (1.77). By next constructing a set {Ok} with angular density A(O) for the index per),

r Fig. 6

94

ENTIRE FUNCTIONS WITH ANGULAR DENSITY

[CHAP. II

and then forming the canonical product with these points, we obtain the required entire function. The construction of the entire function with a given indicator is carried out in § 4 of this chapter. The results may be formulated as in the following theorem. THEOREM 3. 4 For ellery given strong proximate order p*(r) (lim r_<Xl p*(r) = p > 0), and for every trigonometrically convex (with this p) function h(O) of period 21T. there exists an entire fUllction of proximate order p*(r) with indicator p*(O). An analogous statement is true for an arbitrary proximate order if p is a nonintegral number.

We note that the entire function with a given indicator which we shall construct in § 4 possesses a special regularity of growih. When the zeros of an entire function are regularly distributed, one can obtain an asymptotic expression not only for the function In Ij(re i8) I but also for the function argf(z), i.e .• one can obtain a representation for the entire function fez) itself. § 5 of the present chapter is devoted to this topic. In order that the expression argf(z) may have a definite meaning for the canonical product 00

f(z)

= IT a (~; k=l

a"

p)

(p = (p».

we define the function arg G(u;p) in the u plane, cut along the ray (1, +(0), to be zero on the upper side of the cut and then we extend in a continuous manner over the entire plane. The function arg G(u; p) that is defined uniquely in this way over the entire cut plane will have a jump equal to 21T at each point of the ray (\, +(0). We shaH say that a ray arg z = 0 is an "ordinary" ray if it satisfies the condition

. {-I· \1m 1m

h-+O ,.-+00

nCr. O-h. O+h)} () r P ,.

0•

All other rays will be called "exceptional" rays. The monotone nature of the function ~(O) implies that if the zeros of the functions are regularly distributed then the set of exceptional rays is at most denumerable. The basic result of § 5 can be formulated in the following way. THEOREM 4. Ijm is a regularly distributed set with index p(r) and iff(z, m) is its canonical function, then for any z belonging to an ordinary ray, and not belonging to some CO-set, the following asymptotic relation will hold:

In !(re i8 • !Jl) ~ J (6) r p(,,).

(2.08)

• This theorem is a generalization of a theorem of v. Bernstein [I; 31 on functions of normal type for p(r) - p.

Sec. I]

BASIC RESULTS

where

f•

J(6)=_K_ elp('-~-")dA(IjI) sin Kp I-a.. for a non integral p, and J(6)

= -I

f•e1p

(1-+) (6

-~)dA(IjI)+Aelp(8 -10>

9S

(2.09)

(2.10)

8-a..

for a p that is an integer. Here the convergence of r-p(r) Inf(re io ) to the limit J(O) will be uniform in 0 on any closed set IDl of ordinary rays.

We note that the multiplication of the function fez, 9l) by an exponential factor eP(z) in which the exponent P(z) is less than p will not invalidate the asymptotic equation (2.08). The exclusion of the CO-sets in Theorems I, 2, 4 and S is based on the Cartan estimate (see Chapter I, § 7). In this estimate the set of exceptional circles is not effectively constructed. It is for this reason that Theorems I, 2, 4 and S assert only that the asymptotic equations are valid outside of certain exceptional regions (CO-sets) but do not indicate how these regions can be constructed. However, with certain auxiliary limitations on the roots of the functions, one can give more precise information about the position of the circles of the exceptional set. We assume that the points can not come arbitrarily close to each other. More precisely, we assume that one of the following conditions (C) or (C') holds. (C) There exists a number d > 0 such that the circles of radii p (I 11ft I)

Tft=dlanl

1---II

with centers at the points an do not intersect. (C') The points an lie inside angles with a common vertex at the origin but with no other points in common, which are such that if one arranges the points of the set {an} within anyone of these angles in the order of increasing moduli, then for all points which lie inside the same angle it is true that

Iak+1I-1 akl > d Iakl l - P ( 111,,1) for some! d > O. In the conditions (C) and (C') the proximate order p(r) has the same meaning as in condition (A). A regular point set {an} satisfying one of the conditions (C) or (C') will be called a regular, or more briefty an R-set, while the circles Iz - ani <; rn in the first case, and Iz - a,,1 <; d lanl l - PlI••!> in the second case, wiu be called the exceptional circles of the R-set (CKeireles). In § 6 we will prove the following theorem.

96

ENTIRE FUNCTIONS WITH ANGULAR DENSITY

[CHAP. II

THEOREM 5. If the set 9l is an R-set of order per), then outside the exceptional Clrcirc/es the canonical function f(::, 91) satisfies the asymptotic equality

(2.05')

and if arg ::

= 0 is an ordillary ray,

thell

In f (rc i8 , 9l) :::::dn (0) rP (ri,

(2.08')

where H'JI(O) and J'JI(O) are defined hy the equations (2.04), (2.06), (2.09) and (2.10). The conl'erf!,cnce of the functions r- pI,) In Ij(re w, 91)1 and r pI') Inj(reio, 91) to their limits is uniform in the same sense as that of Theorems 1, 2, and 4. In this theorem, the exceptional set is not a CO-set; however, if the condition (C') is satisfied, then the upper linear density of this set can be made arbitrarily small by the proper choice of d. When condition (C) holds, one can make

another "density" arbitrarily small, namely, the upper limit of the ratio of the sum of the areas of the exceptional circles lying in some circle Izl ~ R to the area of this circle arbitrarily small when R -. 00. We shall need the following definition. The relative measure of a set of positive numbers is defined to be the limit I'1m mes (E")

m e(E) -_

ro+oo

r

,

where E' denotes the intersection of the set E with the interval (0, r). If this limit does not exist, then we shall consider the upper and lower relative measures

me (E)

=

lim mes (E''), r

ro+oo

me (E)

=

-

lim mes (E,) • ro+oo

r

A set of zero relative measure will be called an EO-set. It is obvious that the set of those values of r for which the circle Izl = r intersects a CO-set is an £O-set. Hence, in the formulation of Theorems I and 2 one may replace the requirement that z should not take on the values of some CO-set by the requirement that Izi must not take on values from some EO-set. As has been mentioned before, it follows from Theorems 1 and 2 that if the zeros of the function are regularly distributed one can omit from the values of r an £O-set such that the functions of ()

h

f. r

(6)

=

In

If (re's) I rP(r)

will form an equicontinuous family. This property can be extended to arbitrary entire functions of finite order, for which it takes on the following form. THEOREM 6. To every entire function f(z) of proximate order p(r) and to every positive number 1] there corresponds a set E~ of positive numbers with an upper relative measure less than 1] such that if r E £'1' the family offunctions h,.r«() is equicontinuous.

Sec. 2]

ENTIRE FUNCTIONS OF NONINTEGRAL ORDER

97

Theorem 6 can be extended to functions that are holomorphic within some angle 0( < arg z < p. This generalization can be formulated in the following way. THEOREM 7. Let us assume that the function F(z) is holomorphic and of proximate order p(r) within the angle 0( < arg z < p, and that for some positive I < I, for some positive N, and for all sufficiently large valuer of r, every circle

Iz-re contains a point

Zl

i .+~ 2

-a

I
= r1e i81 at which In I F(Zl)/ > _Nr~("ll.

(2.12)

Then for every positive 1/ there exists a set E~ of positive numbers with an upper relative measure less than 'fJ such that if r E ETj' the family offunctions of ()

h,.,.(fJ) = rP("'ln I F(re i8 ) I is equicontinuQUS within the angle a + £ < (J < {3 - £.4 This theorem. which we shall prove in § 7 of the present chapter, is quite useful in the investigation of various questions in the theory of the growth of functions of finite order. 4a

2. Entire Functions of Nonintegral Order with Regular Distribution of Zeros (Proof of Theorem 1) For functions of nonintegral order the exponential factor in front of the canonical product does not affect the indicator of the function. The indicator is completely determined in this case by the canonical product or, what is the same thing, by the set of zeros of tht function. Therefore, we shall consider in this section the canonical product only, which will not restrict the generality of our considerations. Let us first consider some particular cases of regular distribution of the zeros. We have the simplest case when all zeros of the canonical product TI(z) lie on one ray arg z = "P with some density fl. =

.

,,(r)

11m -(") • P

,..,. 00

r

This case has been considered separately in Theorem 25 of Chapter I. Indeed, if one rotates the plane through an angle "P. i.e .• if one substitutes for (J 4 The auxiliary condition given by the inequality (2.12) for the case of an entire function is satisfied in view of a known estimate of an entire function from below (see Chapter I. pages 21-22). 4& A far-reaching development of Theorems 6 and 7 has been obtained by Griiin [~) and KrasiCkov [1; 3).

98

[CHAP. II

ENTIRE FUNCTIONS WITH ANGULAR DENSITY

the quantity () - tp. then for

r, > 0 one obtains the limit i8

. InTI(re)· = -7t~ \ 1m - e ipl9-,!,-1t\ r ~ co r P (T) sin 7tp T

where the convergence is uniform relative to () in (tp Equating the real parts, we find that

+ rt

~ () ~ tp

+ 27T -

'YJ)'

'9

. In I TI (reI) I = -7t~ \ 1m - - COS r, ('J"- ' I'" .- _.). . T~OO r P IT) sin 7tp , ,

The case next in complexity occurs when the zeros of the canonical product II(z) are located on a finite number of rays arg z ='1'; (j = 0, I, ... ,m), and the density of the set of zeros exists on each ray, that is, when we have the limits Ttj (r) ~ r~co r P .

Aj = 11m

(j=O, I, ...• m).

where nir) is the number of zeros of the product II(z) in the interval (0. r) on the ray arg z = tp;. In this case the function II(z) is the product of functions II ;(z) whose roots lie on a single ray arg z = tp;. This case can be reduced to the preceding one. and we have the limits lim

T~OO

In I II (re ffl ) I r P(r)

7t:' = - - ~. A· coso (5 -w.-1t) sin 7tp ~ J ' 'J J~O

where the convergence is uniform relative to () if I() - tp;1 ~ rt > 0 (j = O. I, ...• m). If one writes the sum of the right hand side of this equation by means of a Stieltjes integral, one obtains formula (2.04) for this particular case of regular distribution of zeros. The transition to the general case is based on the following lemma. LEMMA

I.

Let us assume that the set {an} of the zeros of the canonical

product co

II(z) =

IT a (:n; p) 1

has a density with index per), i.e., there exists the limit A _

a -

I'

rt (r) Im-C-)' rPr

and suppose that p = limr-<x; per) is not an integer. Let us denote by II"(z) another canonical product 00

IP (z) =

IT a ( an~ ; p) , 1

in which la~1

= la,,1 and larg a~

- arg ani

< «5.

Then for every

E

> 0 and rt > 0

Sec. 2]

ENTIRE FUNCTIONS OF NONINTEGRAL ORDER

there exists a ()

>

99

°

such that

lin I II(z) I-In I n~(z) II < er W ) for al/ z that do not belong to some exceptional set of circles C with upper linear density less than?J. (Here b depends only on the numbers E and ?J, while the exceptional set of circles C depends on the sets {an} and {a~}.) This lemma makes it possible to replace, with an arbitrary degree of asymptotic precision, a given canonical product by another one the roots of which lie on a finite number of rays. The proof of this lemma is somewhat cumbersome and we shall give it later in order not to interrupt the train of thought. Let us now continue the interrupted proof by making use of the lemma. Thus let us assume that fez) is an entire function of integral order p, and that the set of its zeros has an angular density !:::.,6 with index per). Let the quantity H(fJ) be determined by formula (2.04). We construct the sum

Sm'(~) =

5l:

m-l

~ cos P (6 -lJIj -

1tp

j=O

r.) [A OJ+1) - A (~J)]

(Ym = Yo + 2r.).

It is obvious that for every given positive number E there exists a positive number b such that if max; IVim - Vi;1 < b (j = 0, 1,2, ... , m - I), then (0

-< /j < 27t).

(2.13)

We assume, as has been mentioned above, that the function f(z) is a canonical product co

f(z)

= II a (a:;

p)

(p-I

1

Now we construct another functionp(z) the zeros of which lie on the rays arg z = Vi; (j= 0, I, ... , m - I), and which asymptotically differs little from fez). For this purpose we select the points a~ so that la~1 = la .. 1and arg a~ = Vis, where Vi; <:; arg aft < Vij+l' By Lemma I, the function

will satisfy, for any given the inequality

E

> 0, ?J > 0, and for a

Iln/f(z)/-lnll(z)II<;

rP(r)

sufficiently small positive b, (/zl=r)

(2.14)

for all values of z that do not belong to an exceptional set C of circles whose upper linear density satisfies the inequality p*(C) < "112.

100

ENTIRE FUNCTIONS WITH ANGULAR DENSITY

[CHAP. II

The zeros of the function f6(Z) are regularly distributed, and they lie on a finite number of rays arg = = 'I'i (j = 0, I, ... , m - 1). In such a case, as was indicated above, the quantity r- P ,r) In

It (re il ) I > u.

will converge uniformly to S,,(O) for any u > 0, and 10 - '1';1 words, there exists a number r.,<1 such that if r > r •. a and (j = 0, I, ... , m - I), then

10 -

In other 'I'il u

>

(2.15) From (2.13), (2.14) and (2.15) it follows that if 18 - v';1 1, ... , m - I), re iO E C and r > r.,<1' then

I r-P (r) In If(re il ) 1- H (fJ) 1< B.

> u> 0 (j= 0, (2.1~

The requirement r > rr,,, may be dropped if on; mcludes the circle Izl '" r.,a in the set C. We have proved that the inequality (2.16) holds outside certain angles, i.e., if 10 - '1';1 > u > 0. In order to free ourselves from this auxiliary requirement, we shall make a second subdivision of the interval (0, 217 + 15) by means of pointsV'; (j = 0, I, ... , m - I, m) such thatV':" = 'I'~ + 217, and 1'1'1+1 -' 'I'jl < 15, and by selecting u > 0 so that the sets 10 - '1';1 u and 10 - '1';1 u (j= 0, I, ... , m - I, m) will cover the interval [0,217] completely. Making use of this second subdivision, we find that the inequality (2.16) is valid if re iO does not belong to a certain exceptional set C' with upper linear density p*(C') < 'YJ/2, and if 10 - 'I'jl > u (j= 0, I, ... , m). Because the sets 10 - 'I'il u and 10 - 'I'il u cover the entire interval [0, 217], we see that the inequality (2.16) is satisfied when re iO E C = C + C', where r(C) < 'YJ. One can thus construct an exceptional set C of circles, with an arbitrarily small upper linear density, outside of which the inequality (2.16) will be satisfied for any preassigned positive E. Making use of this fact, we shall construct a set of circles, of zero linear density, outside of which set the required asymptotic equation will hold. For this purpose, we select two arbitrary null sequences of positive numbers {e,l} and {'YJ,I} and construct a sequence of sets C ,I (p = I, 2, ...) of circles such that p*(C,,) < 'YJ,I and that, for reie E C ,I,

>

>

>

>

I'-P "'lin It(re i9 ) 1- H (IJ) i

< Bp.

(2.17)

We shall denote by l:r rf the sum of the radii of those circles of the set C,I whose centers lie in the circle Izi < r. Since C ,,) < 'YJ", it follows that for r sufficiently large we have the inequality

r(

~r~

< 2"1,1".

(2.18)

Sec. 2]

ENTIRE FUNCTIONS OF NONINTEGRAL ORDER

101

Let us select a sequence of positive numbers R., (p = 1,2, ... ) such that R1J+l > (p + I)R" and that the inequality (2.18) is satisfied for r > R". From each set C" we pick out those circles whose centers lie in the ring

Rp" Izl < Rp+1' and we denote the set of these circles by C. This set is a CO-set. Indeed, if we reorder the circles of C and denote by r; the radius of the jth circle, we find that for we have the inequality

.!.r~ ~r rj < 2.;"11... p +3. :"1:1... p + ... + 2'11p < 2"1tp e + 21Jp • Therefore,

p*(C) = O.

Furthermore, if reiO E C, and if r >~, the inequality (2.17) is satisfied. In other words, if Z E C, then the quantity

hr, r(fJ) = rP (f') In I/(re i ') I converges uniformly to H«(). This completes the proof of the theorem.' We shall now prove Lemma 1. I. Let the positive numbers (j, T and {J be given such that (j < I, T > 1, and {J does not exceed either I - (j or 7' - 1. We shall represent the infinite product co

IT a (a:; p)

I(z) =

1

as a product of the following functions:

1o (z)=

IT a (a:; p), lo,~(z)= II

II a (a:; p),

./(z)=

IlInl
w< Ian I < (t-~) r

IlInl>u-

a (aZn ;

p),

• It is useful to note still another formula for the function H(8),

f

2K

H(6)

= 5-I~ nap

cos P (16-

"'1-K)d~<4).

which can be obtained directly from (2.04) if one notices that if 8 - 2" < " < 8, then cos p(8 - " - ,,) = cos P(18 - "I - ,,) and cos P{18 - (" + 2,,)1 - n} = cos P(18 - '" - ,,).

102

ENTIRE FUNCTIONS WITH ANGULAR DENSITY

[CHAP. "

In the sequel we shall consider, in addition to these functions, also those functions which we obtain if we replace in each product the points an by points a~ such that lanl = la~1 and larg an - arg a~1 0). For these new functions we shall retain the earlier notation except for the superscript b (for example,p(z),f!(z), and so on). We have the inequality lIn II (z) I-In

Il (z) II

<: lin: la (z) I-In II! (z) II + lin I~I (z) I-In IJ Ii (z) II +1 In Ila. ~(z) I-In 11;.~(z) II + lin k ~/(z) I-In k ~l(z) II

+ lin I ~~ (z) I-In I ~: (z) II.

(2.19)

Let us next estimate the value of each of the terms on the right hand side of this inequality. 2. From Lemmas 7 and 8 of Chapter I, it follows that for sufficiently small (1 > 0 and for sufficiently large" > I we have the asymptotic relations lin

Ila (z)11 < rorP(r).

lin 1,/(z)l/

< rorP(r),

lin

II: (z) II < rorP(r),

lin I,l(z) I I

< iorP(r)

and hence,

I In Ila(z)I-ln I/!(z) II < i- rP (r), Ilnl./(z)I-lnIJ5(z)11

< irP(r).

(2.20)

3. To estimate the third and fourth term we shall make use of the uniform continuity of the function

in sand

In I 0 (se- i9 ; p) I

°

(s>O)

within the regions

(1 -

p)-1

-< s -< a-I,

.. -1

-< -< (1 + ~)-I. S

From the existence of the density 6. of the set of zeros {an} (a" ~ 0) it follows that for some c > 6. n (r)

< crP

If we select 15 > 0 so small that for (1 - {f)-I and for IOl - 02 1 < 15 the inequality

(2.21)

(r) •

< s < 0--1, or ,,-1 .;;

lin 10 (se i81 ; p) I-In I 0 (se iS.; p) II

< l~~P

S

<:; (I

+ {f)-I,

Sec. 2]

103

ENTIRE FUNCTIONS OF NON INTEGRAL ORDER

is satisfied, then we obtain lIn Ila, ~ (z) I-In I I!. ~ (z) II

-<

~

ar< I an I «1-,),.

Ilnla(a:;p)\-lnlaC~;p)ll
II lIn Ia (a:; p) \-In Ia L~ ; p) t\
lIn I~, ~/(z) I-In ~.pl (z)

~

<

(1+P)r< I tJ,a1 <..T

"

From (2.21) it can easily be seen that for sufficiently large r 1l ('tr)

< 2c,P rP(TI

and the asymptotic inequalities hold: lIn Ila,p(z)I-ln

II!. ~(z) II < ;

lIn I., /(z) I-In I~, ~l (z)11

l rP(r ,

}

(2.22)

<;

rP(rl •

It remains to show that for a small enough positive Pand for Z not belonging to a certain exceptional set, the last term in (2.19) is asymptotically less than er P(TI/5. This is the most difficult step of the proof. 4. In place of the function /J(z), we first consider the function

YB"

II

(z) =

I1-PIB<1

(1-

an I < (1+~IB

a:)

(o<~<;, R>O)

(2.23)

and estimate lin I'PR,/J(z)11 in the ring IIzl - RI < PRo Here we do not make immediate use of the existence of the density of the point set {a,,}, but only assume that the set has a limit point at infinity. Thus we obtain a general estimate (see (2.27», and then we utilize the density and obtain the required inequality. First of aU let us find an upper bound for the function I'PB,/J(z)l. From the inequality

11 which is true when IIzl - RI < PR, then

:"1 -< (1 + I;" I) -<, 2 p

(r = Izl),

Ila,,1 - RI <; PR and Ir - RI <; PR, it foUows that if In I~R,p(z)1

< N(R.

~)In

2

I_P'

(2.24)

Here N(R, p) denotes the number of points of the set {a .. } that lie in the ring (I - P)R <; Izi < (1 + p)R.

104

ENTIRE FUNCTIONS WITH ANGULAR DENSITY

[CHAP. II

In order to obtain a lower bound of tpR.P(Z), we start with the Cartan estimate (Chapter I, § 7) according to which for every H > 0 the polynomial

II

P(z) =

(z-a n )

(t-~)R
R

satisfies the ineq uality InIP(z)I>N(R,

~)In(~)

everywhere outside circles the sum of whose radii is equal to 2H. Noting that in the product (2.23) la.. 1< (I + P)R and setting H = {PR, we deduce from this inequality that outside the exceptional circles with the sum of radii equal to 2{PR the following inequality holds: ~11

In "1R, ~(z) I> N(R, ~) In e(l + ~)

.

Let us now construct the exceptional set C in the entire plane. For this purpose we divide the positive half-line into semi-intervals Ii'

Rj (l - ~)

-< , < R

j

(l +~)

(j=O, 1,2 •... ).

where

Rj

+

I ~)J = ( I-~ ,

while the part of the plane outside the circle Izl ..;;;; I - P is divided into the rings K i • (j = 0, 1, 2•... ). RJ(I-~)-< Iz I RJ(I +~) We shall denote by n; the number of the points a.. lying in the ring K;. With each ring K; one can associate the exceptional circles. with sum of radii equal to 2{PR;. outside of which the following inequality holds:

-<

In 19RJ, ~ (z)1

> nJ In e(/~~) > nJ (21n ~ -In 2-1).

(2.25)

Since each exceptional circle contains a point a .. from the ring K i • the centers of these circles lie either in KI or in one of the adjacent rings KI-l and Kl+l' Let us denote by C the set of aU exceptional circles that correspond to the various rings XI (j = 1,2•...) and by T" (k = 1,2, ...) the radii of these circles. We now shall give an estimate of the upper linear density of this set. Let R be an arbitrary number greater than 1 - p, and let m be such R.. < R < R..+1' Then

~R'k

",+2

-< J=O ~ 2~'!RJ < ~ (I +~) Rm+2 -< ~ (I + ~)S(I -

~)-2 R.

On the left side of this inequality stands the sum of the radii of those circles of C whose centers lie in the circle Izl ..;;;; R. Therefore,

p·(C)

-< ~ (1 + ~)S(l- ~)-2.

Sec. 2]

105

ENTIRE FUNCTIONS OF NONINTEGRAL ORDER

The inequality (2.25) yields a lower bound of the function I'f'RjZ)1 inside the circle Ki but outside the set C which we have constructed. Let us now proceed to estimate the function I'PR,iZ) I outside of this set for an arbitrary R. For this purpose we express the function 'PR.P(Z), for R.,. <; R <; R.,.+1' in the form

where

II (1 -~), 'f~)~(Z)= II < (l-~)R (1 -~). (l-~)R," y~) ~(z) = •

(l+~)B<:lanl«lH)R,"+l

an

<: I an I

•

an

The functions 'PR"..f3(z) and 'PRtlH1.p(Z) are estimated by means of the inequality (2.25). The functions 'Pk~1(z) and 'f'~~1(z) have to be estimated from above, i.e., we have to find upper bounds for them. In order to do this we note that in each one of these functions

lanl > Rm(l-~)'~.R(l-~»)(I +~)-1. and hence for Izi

< (1 + {3)R

II Therefore, if Izl

I + I~ < 1 + (l(\ + ~)2 < 10 . alii -

~ ~I an ""

~)2

< (I + {3)R

In I y~~ ~ (z)1

< nm+1In 10,

In I y~~ ~ (z)1

< nm In 10.

Making use of the bounds of all four of the functions, we obtain, when

Ilzl - RI < {3R and z lies outside the circles of the set C, the inequality In IYR.~(Z) I> (n m +nm+1) [21n ~ -

1 -In 10).

Combining this result with (2.24) we see that lin I YR. ~ (z)

+

II < (n", + nm+1) [2 In + 4] .

(2.26)

Let us now find bounds for the function

4>R.~(Z)=

II

(l-~)R<:lanl«1+~IR

Q(~; all

p).

It satisfies the equation In 14>R. ~ (z)1 = In I'~R. ~ (z)1

+Re{

~

(~+ ~: + '" + ZPp)}.

(1-Ii)B<1 Gnl «1+~)B 4 n "

pa"

106

ENTIRE FUNCTIONS WITH ANGULAR DENSITY

[CHAP. II

A bound for the first summand is given by the inequality (2.26), while for an estimate of the second term one needs only to note that if (I - {J)R ~ an < (I + {J)R and IIzl - RI < {JR, then

~1<1+~ 1 an 1-~ and hence

~

ICz +;a

12

(1-~)B", lanl «1+~)B

n

n

+ ... + P::)1 n

+~)p f.l / ' P

Combining these bounds we find that if z E C and

Ilzl - RI < {JR, then

+

lIn I II»B. ~(z)11 < (nm + nm+1) [21n + BpJ,

(2.27)

where

Bp = 4+3 Pp. 5. We now make use of the fact that in our case the set has a density, i.e., the limIt 1

=

lim n (r) r-+ 00 rP (r)

exists. If R",/(I - f1) " R " Rm+ 1/(1 - f1) and

Ir -

R I " f1R, we have

and lim R -+ 00

n"';1_< r P\ )

lim !II

nm;! ) = (11 + P)P [(I + ~)P - P

-+ 00 R~II( m

(I -

~ll~.

From these inequalities and (2.27) we obtain the asymptotic inequality

:r

I In III»B. ~ (z)11 < 2( 2 In f-+Bp) G+ [(I+~t -(I-~tl ~rp(r), in which Ilzl - RI < {JR and z E c. It is now obvious that for any given pair of positive numbers ~ and , one can find a positive number {J and a set C of exceptional circles such that p*(C) < ~ and that outside these circles

IIn 1II»B. p(z)11 < ~rP

(r)

(I z 1=

r).

6. Let us select ~ = nl2 and, = e/lO, and construct the sets of circles C1 and q corresponding to the sequences {all} and {a~} of Lemma 1. Then for

Sec.2J

107

ENTIRE FUNCTIONS OF NONINTEGRAL ORDER

R = r = Izl, for sufficiently small {J, and for any asymptotic relations

z E C1 + C~,

we have the

IIn I ~:. ~ (z)11 < Th rP (,.) and hence

Ilnl~,..~(z)I-lnl~~:~(z)11

<~

rP(r).

Furthermore,

p·(C) < '1). Thus we have obtained the required bound for the last summand in the inequality (2.19), and we have completed the proof of Lemma 1. This establishes Theorem 1. In the proof of Lemma 1, subsections 4 and 5, we established two propositions which we shall use in the sequel. For this reason we restate them here as separate lemmas. LEMMA 2. Suppose that {aft} is a sequence of complex numbers with a single limit point at infinity, and let

~B.~(Z)=

II

(l-~)B'" lanl «1+~)B

where p is a natural number, 0

a (aZ ; p).

< {J < i, and R > 1 R

j

(2.28)

n

{J. Let

=(1I-p +~)i

and nj be the number of points of {aft} which lie in the ring (l-~)Rj

<. I zl < (1 +~)Rj'

Under the above conditions one can select a set C of circles, with an upper linear density ~·(C) ~ (1 ~)s(l _ ~)-2.

-<

such that ijz E C,

+

Ilzl - RI < {JR and Rm <; R <; Rm+b

then

IIn I ~B. ~(z)11 < (n m + n''I+1) (2 In ~ +Bp), where B" = 4

+ 3"'p.

LEMMA 3. If the sequence {aft} of points of the complex plane has density t:. with index p(r), then for every pair of given positive numbers E and, one can select a set C of circles, with upper linear density less than E, and a number {J such that if R - 00, Ilzl - RI < (JR, and z E C, then the function ~B.~(Z) given by (2.28) will satisfy the asymptotic inequality

IIn I B. p (z)11 < J? (r)

108

ENTIRE FUNCTIONS WITH ANGULAR DENSITY

[CHAP. II

We call attention to the fact that in Lemma 2 we did not impose any restrictions on the sequence {an}, while in Lemma 3 we required only the existence of the density. It was unessential whether p = limr_c:c per) was an integer or not. 3. EDtire FUDdioDS of IDtegral Order with Regular DistributioD of Zeros (Proof of Theorem 2)

For functions of integral order the indicator is not completely determined by the set {a .. } of the zeros of the function, but may depend also on the exponential factor which stands in front of the canonical product. Furthermore, in this case, for a given proximate order per) the quantity t1" as was shown in § 13 of Chapter I, does not determine the type of the canonical product. For us it is important that in the case of an integral order the considerations of the preceding section do not carryover even for the simplest case when all the zeros of the functionj(z) lie on a single ray. The fact was already mentioned in the proof of Theorem 25, Chapter I, § 17. In order to obtain an asymptotic equation in this case, we express the functionj(z) (as was done in § II, Chapter I) in the form (2.29) where

I,. (z) = Here, P rl(Z)

II a (a p- 1) I II> ,. a (a p). Z

I an I ..:: ,.

Z

;

an I

n

;

11

= Co + c1z + ... + CrlZ-1 and

o,er) = cp + f ~ a;;P. I all I <,.

If the zeros an and the quantity the existence of the limit

Cp

satisfy the condition (B), i.e., if we have

1" ( ) ' \1m -L() 0, r

r-+oo

r

='t,e -ip8t,

then the quantity l(r,

6)=,-~(r) (qlnz+Pp_l(z)+~f(r)zp]

will converge uniformly in

(J

(0 <;

(J

<; 21T) to the limit

lim 1 (r, 6) =

't,ei p (8-8,',

(2.30)

"-+00

This circumstance explains the meaning of the condition (B). The entire problem is now reduced to the study of the second factor f,,(z), for which we shall need only the existence ofthe angular density of the set of zeros, As before,

Sec. 3]

109

ENTIRE FUNCTIONS OF INTEGRAL ORDER

let us consider first the simple case when all the sets {an} that possess angular densities are located on a finite number of rays arg z = "Pj (j = 0, I, ... , m - I). Now applying Lemma 9, § 17 of Chapter I, instead of Theorem 25, we find that if 10 - "Pjl ;> (1 > 0, lim In II,. (rel~) I = -

,.-+00

r P(")

11/-1

~ ~j

~

~j

[(fJ -

-

9j) + ...!.. cos P (fJ - '~J)]

'it) sin p (fJ -

p

j=O

,

where the convergence is uniform and ti j stands for the density of the set of zeros on the ray arg z = "Pj. One might expect that also in the general case of the existence of an angular density of the set of zeros outside some exceptional set one would have the equation

o

=

J

[(6 -

~-r.) sinp (fJ -9) -

+

cosp(fJ -9)]

d~(~).

0-2"

In order to prove that this is actually the case we need a lemma which is analogous to Lemma I. LEMMA

4.

If p is an iI.teger, if the set f,.(z)=

II

I an 1<:"

{a,,} of zeros of the function

o(~; P-l) an

11> "0 (:; p)

I an I

n

has a density with index p(r) - p, and if

f;(z)=

II 0(;,; p-I) IT I a' I <: T n

n

0C~;

I a' I > " n

r),

n

where la~1 = lanl and larg a~ - arg a,,1 < lJ, then for every 7J there exists a lJ > 0 such that we have the asymptotic relation

>0

and

E

>

°

lin If~(z) I-In IJ,,(z) II < erP (t"1 for all z not belonging to a certain exceptional set C of circles with upper linear density less than "I. (Here lJ depends only on E and "I, while the exceptional set C depends only on the sets {all} and {a~}.)6& The proof of this lemma is analogous to the proof of Lemma 1, but it can be considerably shortened because we use the established Lemma 3. We give the proof here. '" Lemmas I and 3 of this c:hapter have been developed further by KrasiCkov [1; 4). See also Gol'dberg (5).

110

ENTIRE FUNCTIONS WITH ANGULAR DENSITY

[CHAP. II

Setting

I,.." (z) =

II

a (: ; i

J -

1an I": ar

·1r. ~ (z) =

exp

1)·

(0

< a < 1).

n

i

z> .. ~ I "11 1< ,i ....':r

we obtain

lin II,.(z) I-In II:'(z) II -< /In II,., .. (z) I-In II;... (z) 1/ + lin IJr(z) I-In IJ;(z) I I + lin IIr, a, (i (z) I-In I ~ (z) 1/ + lin I., ~/(z) I-In I., ~f' (z) II + lin I 4> r, ~ (z) I-In I 4>:. ~ (z) II + /In I '~r.~(z) I-In I ~:. ~(z) II· (2.31)

I; ...,

From Lemmas 7 and 8 of Chapter I it follows that for any given E > 0, for a sufficiently small (J > 0, and for a sufficiently large T ;> I, we have the asymptotic relations

lIn II,., .. (z) II < lIn I./r (z) II <

}'2

;2

rP

rP

(,.1,

(r).

/lnl.t:.,,(z)I/<

}'2

lIn I•.t:(z) 11<

;2

rP(r),

rP (r).

Therefore

IInl/r... (z)I-ln I.t:... (z) II<

~

lIn I .I,.(z) I-In I •.t:
i

rP(rl • )

(2.32)

rP (rl.

Sec. 3]

III

ENTIRE FUNCTIONS OF INTEGRAL ORDER

From the existence of the density ..l of the set of roots {an} of the function > ..l

!(z) it follows that for some c

n (r)

< cr P \r)

(r

> 0).

Making use of the continuity of the functions In

10 (se- iS ;

1) I

p-

(1 -

if

~)-1 "" S < 0- 1

and InIO(se- i8 ; p)1

if

't-l<S<(1+~)-t.

I

we can (just as in subsection 3 of § 2 of this chapter) select a positive 15 so small that

lin I Ir. ~ (z) 1- In i I;. ~ (z) i I < i- r~ (rl, G,

G.

lin I tlr. ~ (z) I-In I.I;, ~ (z) II <

i-

rP

(2.33)

(r).

By Lemma 3 one can choose a number {3 (0 < {3 < 1) and a set C of circles, with upper linear density less than the given number?} > 0, such that if:: E C then

lin I r,~ (z) II <

112 r P (rl,

lin I ~, ~ (z) II <

;2 rP\rl,

i.e.,

Ilnlr.~(Z)I-lnl~.~(z)II<~

(2.34)

rP(r).

Furthermore, 1

lin I ~r.~(z) II 0).

> 0 sufficiently small, \In I ~r.~(z) I-In 14~.~ (z)11 < ~ rP(r).

(2.35)

From the inequalities (2.31), (2.32), (2.33), (2.34) and (2.35) it follows that for a sufficiently small positive 15 and for z outside of the set C we have the inequality lin I Ir(z) I-In II~(z)

II <

IfP (rl.

This proves the lemma. Let us now consider the asymptotic representation of the function In Ifr(z)l. Such a representation is given by the next lemma.

112

ENTIRE FUNCTIONS WITH ANGULAR DENSITY

LEMMA 5. If the point set {an} has angular density where p is an integer, and if

II

I,(z)=

a (: ; P-I)

I an I " ,

n

II

0

I an I > "

[CHAP. II

~(1p)

with index per),

p),

(~; a,.

then for any z exterior to a certain CO-set the following asymptotic equation is mlid In II" (re ie ) I ~ Hi (6) rP (r), in which

H.(fJ)

=-

PROOF.

(9

~-2x

[(6-y-1t)Sinp(6-y)

+ ~COSP(6-4)]dt::.(,~). p

For the integral H 1(O) we construct the sum

S.(6) =

•

- ~[(6-'~J-1t)Sinp(6-oJiJ) J=O

+

40 < '~1 < ...

+COSP(~-'M] [t::.(4J+1)-t::.('~J)I,

< Yn < oJin+l' < 8 U::::::: 0,

I, 2, ... , n), Here we choose the positive number b so small that oJiJ+l -'}J

oJin+1 = ,~o+ 21t.

IHl (6)-Sn(6) 1< i·

(2.36)

Next we select the numbers ai so that 10~1 = lOti, arg oi =

1p;

if 1p; <; arg at

<

'PHI' and we construct lhe function

J: (z) =

U

0 (:, ;

I ak I " " k

p- 1)

11>"

0 (:, ;

Iak

By Lemma 4, for every given pair of numbers e sufficiently small positive 15, the inequality

lIn I/,,(z) I-In If; (z) 1/ < ;

p).

k

> 0, 'fJ > 0,

rP(")

and for a

(2.37)

will be satisfied by all values of z which do not belong to an exceptional set C of circles with upper linear density less than TJ/2. For the function F,(z) it was established that if (1 > 0 and larg z - 1p;1 ;> (1 (j = 0, I, ... ,n), one can find a number r •. a such that for r ;> r •. a

I ,-p (,,) In / I; (re ie) 1- Sn (6) I < ; .

(2.38)

From (2.36), (2.37) and (2.38) it follows that if z E C, and if larg z - 'P;I ;>

(1

(j = 0, 1,2, ... , II), then

I r-' (,,) In /1" (re ll) I -Hl (6) I < a.

(2.39)

Sec. 3]

113

ENTIRE FUNCTIONS OF INTEGRAL ORDER

After making a second subdivision of the interval (0, 211" + 15) by means of the points 'Pi (j = 0, I, 2, ... ,17) in such a manner that 'P~ = 'P~ + 211", 1'I}i + I - 11'1/ < 15, and after choosing a positive a sufficiently small, we get the result that the sets 10 - 'Pil ;> a and 10 - 'Pi I > a (j = 0, I, ... , n) cover the entire interval [0, 211"). Making use of this second subdivision, we find that the inequality (2.39) is satisfied if z = reiO does not lie in a certain set C' of circles whose upper density is less than "1/2 and if 10 - 11';1 ;> a (j = 0, I, ... ,n). In this manner the inequality (2.39) will hold everywhere except for a set of circles C = C + C', where

p.(C) < '1\. In exactly the same way as in § 2 for the proof of Theorem 1 (pages 100-101), one next constructs the CO-set outside of which the relation (2.40) holds uniformly relative to O. We note once more that in this derivation we have made use only of the existence of the angular density of the set of roots. Lemma 5 has thus been proved. Let us now proceed to the proof of Theorem 2. By hypothesis, the point set {a,,} satisfies the condition (B) (see page 91). If one makes use of this condition, the representation (2.29) and the equations (2.30) and (2.40) yield at once for z E CO the uniform convergence of the limit

11m In I/(r~) I r

-+ 00

,.,

(r)

= H (6),

(2.05)

where

H(6)

= 't,cosp(6- 6,)

- f•

[(6- +-1t)sinp(6-+>++coSP(fJ-+>JdA(+>.

I-a.

Let us transform this expression to the form (2.06). For this purpose we note that for an integer

• f [! cosp(fJ -+>+1t sinp(6- +>] dA(.~)

I-I..

a.

=

JI! COSP(6-+)+1tsinp(6-+)]dA('~) = AcospfJ+B sinp8.

114

ENTIRE FUNCTIONS WITH ANGULAR DENSITY

[CHAP. II

where A and B are constants, and we find that 8

f (6 -I~)sln p(6 -~)

H (6) = Al cosp6 +81 sin p6 +

dll.(,?).

8-llr<

As was shown at the end of § 16, Chapter I, this inequality implies

A(~)=

[HI(~)+p2

2!p

f+.+

H(tp)dtp] ,

and in view of the periodicity of the function H«(J)7 2"

f

e'?4 dll. (,~)

°

= o.

Taking this into consideration, we find that A

(2.41)

=

B

= 0 and

that

8

H(6) =

't,COiP

(6 -

6,) -

f (6 -~) sinp (6 -

~)d:l(~).

8-2"

This completes the proof of Theorem 2. In § I we stated that if the zeros of the function fez) are regularly distributed its indicator coincides with the function H(O) defined by the formulas (2.04) and (2.06). Let us examine this in greater detail. By Theorems 1 and 2, in the exterior of the exceptional CO-set we have uniform convergence of the quantity ht,T(O) to the limit H(O). We shall show that in the interior of the exceptional circles we have the asymptotic relation (I! 0).

>

Indeed, the ratio of the diameter do of an arbitrary region n, consisting of exceptional circles of the CO-set, to the distance of this region from the origin tends to zero when this distance is increased. From the maximum principle it follows that the value of the function In 1/(z)1 at any point of such an exceptional region does not exceed its maximum on the boundary of this region. But on the boundary of the exceptional region the asymptotic formula for h, .•(O) is valid, and, taking into consideration the continuity of the function H(O), we find that for re i8 E n the following asymptotic inequality applies: In If (re 48 )

1< (H(6) + ; )(r+dg)p(r+dg ) < (H(6)+e)rp(r).

On each ray arg:: = 0 there are points arbitrarily far away from the origin which are not covered by the circles of the exceptional CO-set, and hence for every 0 (18) lim Inll(re )1 =H(6). r-+oo r P (r) 7

The equation (2.41) can be obtained directly from the condition (B).

Sec. 3]

115

ENTIRE FUNCTIONS OF INTEGRAL ORDER

It is interesting to note that the regions n consisting of exceptional circles and not containing zeros of the function j~z) can be omitted from the exceptional set. Indeed, applying the minimum principle to the function In If
h,(6) = Acosp6+Bsinp6

(2.42)

ot <; 0 <; P; A and B are constants). If, however, inside this angle there are no zeros of the function, then for

(here ot

< (J < Pthere exists the limit

h (6) = Um In If(reie) ! ,

r P,,.)

1'-+00

where the variable tends to the limit uniformly when

,

ot

+ "1 <; 0 <; P-

"1 ("1

> 0).

The second assertion follows directly from the fact that under the given hypothesis there are within the angle (ot + "1, P- "1) no points belonging to the exceptional circles. The truth of the first assertion can be deduced from formulas (2.04) and (2.06). For example, formula (2.06) takes the form h(6)

=

f-~ sinp(O -

7T)dA(~)+'Tr:os p(O - Of)·

P-2"

Expanding the sine of the difference of two angles and performing the integration, we obtain formula (2.42). It is useful to consider also the following example of an entire function of integral order with regularly distributed zeros. Let {a .. } be a set of angular density .1(,,) within some angle ot < " < ot + 1T/p, let p be an integer, and let

~ (z) =

J;I ( 00

1 --

z" )

a~P

•

It is obvious that the set of zeros of the function cI>(z) possesses an angular density and that (JI = O. The formula for H(I>(O) can easily be obtained if one applies Theorem I to the function cI>(Zl/2p ) of fractional order. After some simple transformations one then obtains

.

-+-p H(8)=1t

f.

sinplfJ-~ldA(~) .

116

ENTIRE FUNCTIONS WITH ANGULAR DENSITY

[CHAP. II

4. CODStruction of aD Entire FunctioD with a GiveD Indicator (Proof of Theorem 3) Theorem 1 shows that it is possible to construct an entire function of nonintegral proximate order per) (p > 0) with an indicator that coincides with a given trigonometrically convex (for the given p) function h(O) of period 211". Theorem 2 yields an analogous result for an integer p and for an arbitrarily strong proximate order. We shall first carry out this construction for the simpler case of an entire function of nonintegral order. 1. First of all we construct a nondecreasing function a,¥, for a given trigonometrically convex function h(O):

a(l/I) = 2!P [ h'(1/1 + 0) - h'(1/1 - 0) + p2 fot/lh(q» dcp

l

By Theorem 24, Chapter J, the function h(O) can be expressed in the form 8

h (0) =

51:7tp r cos p(O -

'f -. 'It) d£\ ('f).

(2,43)

8:2"

In view of Theorem I, the problem can be reduced to the construction of a point set {an} with an angular density a('I') for the index per). After this set is constructed it is sufficient to write down the canonical product ex>

f(z)

= IT a(-=-· 1 ak '

p)

(p= (p».

By Theorem I, the indicator of this function can be expressed as

f

8

7t _ h,(6)=_I_

s n 7tp

cosp{6-1JI-'lt)d£\{,-\,)

8-2"

and hence h,(O) = h(O). 2. In order to construct the point set {ak } with angular density a('I'), we approximate the function a{'I') uniformly by means of a sequence of piece-wise constant functions am'¥' with a finite number of discontinuities, selecting these functions so that the magnitudes of the jumps am.; (j = 1,2, ... , sm) are rational numbers. The points of discontinuity we denote by 'I'm.; (j = 1, 2, ... , s,,,). It is not difficult to construct point sets 9l", with angular densities .lm('I')' For this purpose we carry out the following operations. We select the values r. = r m " (p = I, 2, ... ) for which the quantity rP(r) takes on values which are multiples of the common denominator d", of all the rational numbers am.; (j = I, 2, ... , s",). After that we mark the points that lie at the intersections of the circles Izl = r "'." (p = I, 2, ... ) with the rays

Sec. 4]

117

CONSTRUCTION OF AN ENTIRE FUNCTION

arg z = 'Pm,; (j = I, 2, ... ,sm)' and we assign to each marked point the multiplicity qm,; = dmtlm,; (j = 1,2, ... ,sm). It is obvious that the number n",(r, 1/1) of marked points within the sector Izl ..;;;; r, 0 < arg z <; 'P is given (with multiplicity taken into account) by n".(r,

,~) = [r~(r)] d". ~ ".

l!"..1

0<+/,+

= ~(,~) rP (r) + c (r.

1\').

(2.44)

where

(2.45) From this it follows, of course, that the constructed set possesses the angular density um('P). Let us now divide the entire plane into rings K m ,

r"'-1 < I z I <: r".. selecting for this purpose a sequence of numbers ro < r 1 < r 2 < ... such that ro = 0 and r m - 00 as m - 00. In each ring Km we select the points of 91 m. The union of all these point sets we denote by 91. We shall prove that if the chosen sequence rm (m = 0, I, ... ) increases sufficiently fast, then the set 91 will have angular density tl('P). In fact, let us select the sequence of numbers r m (m = 1,2, .. ,) such that

(2.46) Denoting by n(r, 'P) the number of points of the set 91 that lie in the corresponding sector, we shall have, for rm < r < rm+l and for some integer T < m, the equation tit

nCr, I\')=n(r",

1\')+ ~In"(r,,, I\')-n,,(rk_l' 1\')] ~+1

+nm+l (r,

For any given

E

1\')-n"'+1 (r.. +).

(2.47)

> 0 one can choose the number T such that for k > T (0

<: I\' < 21t).

Making use of the equation (2.44) and of the inequality (2,45) we find that for k > T n"(r,,, 1\')-n,,(r"_I' ,~)< 1L\(I\')+.)(rP(rk) - rP(rk-l' " It-I From this and from (2.47) we obtain

+2dll ).

.. +1

and, in view of (2.46), -\.-n(r. IjI) /' A(,)+ 1m ..lItr) ~ '-A." ••

r .. co r·

118

ENTIRE FUNCTIONS WITH ANGULAR DENSITY

[CHAP. II

Analogously we obtain . . . . 04(') . n (r, <jI) ..,;?~ I1m 't -E,

~

and, since

E

r Plr )

is arbitrary, we see that

lim n (r, <jI) r -+ 00 rP Ir)

= tl (1jI).

Thus we have constructed a set with the given angular density and have proved Theorem 3 for the case of non integral order. We call attention to the fact that the functionj(z) which we have constructed not only possesses the given indicator h(O) but also satisfies the asymptotic equation

In If(re i &) I ~h(6)r plr) in the complement of some CO-set. 3. The case of integral order is considerably more complicated. In this case it is not sufficient to construct a point set {an} with the given angular density

tl(,~)=

2!P

f

[h'('~+O)-h'(+O)+p2 '"

h(IP) dIP]. (2.48) o It is necessary to construct the set so that the condition (B) is satisfied. Furthermore, comparing the expression (1.82) for the given function h(O) with the expression (2.06) of the indicator, we see that for some cp the limit

'Cr8 -ip9f

= r-+oo lim L (1 ) {cp +.!. r P

~ ai' }

(2.49)

la" I
must be equal to the given quantity. Namely. the following equation will have to be satisfied:

'Cfeos (6 - 6,) = A cos p6

+ B sin p6,

where A and B have the same meaning as in formula (1.82). The canonical function of this set with the indicated value cp will satisfy the asymptotic equation In I f(re i8 ) I~ h (6) r' (r). We shall construct such a set for the given strong proximate order p*(r).7a 4. First we will construct for h(O), with integral p and an arbitrary proximate order p(r), a set {at} with angular density ~(1JI) given by equation (2.48) and such that ~ ai' =0. la" I
Sec. 4]

CONSTRUCTION OF AN ENTIRE FUNCTION

119

For this construction we make use of the fact (established in Theorem 24, Chapter J) that the function ~(1p) given by equation (2.48) satisfies the condition 2ft

f

e-lpejl

dA (~) = 0

o

if p is an integer. Next we approximate the function ~(1p) with a sequence of piecewise constant nondecreasing functions ~m(1p) with a finite number of rational jumps ilm.i (j = 1,2, ... , s) in such a way that these functions satisfy the condition 2ft

f

e-ipejl

dt::.m (~) =

o.

(2.50)

°

The possibility of making such an approximation was demonstrated at the end of § 19, Chapter I. With the aid of the function il m (1p) we construct the set 91 (see subsection 2). This set will have angular density il(1p). Also the points ak will be located on concentric circles Izl = 'm.'P' they will have the form, m.'P eitp"'i (j = 1,2, ... ,sm), and they will have the multiplicity qmi = dmil m; (j = I, 2, ... ,sm) where il mi is the jump of the function il m (1p) at the )oint 1p;. Taking the sum Om. p = ~ai/ over all points which lie on a single circle, with every point a k being repeated according to its multiplicity, we obtain 8m

Om. p = dm';',~p ~ e-ip<\Imi t::.mj = O.

i=1 This equality follows from (2.50), from which it is obvious that

~ akP=O.

la"l
For the construction of a set satisfying for some C p the condition (2.49), and for given T f and Of' one must add to the set 91 some set 91' with zero angular density and satisfying for some cp the condition (2.49). The union set 91 + 91' will fulfill all requirements. It is obvious that a function j(z) that satisfies in . the complement of some CO-set the asymptotic equation In If (re iO ) I~ h (~) rP (r)

is equal to the product of the functions

f(z.!R)=llo(: ;p) "=1 k

(2.51)

z e CPzp n"'· 0 (b; P)

(2.52)

and q> (Z)

=

lJ-~

"

120

ENTIRE FUNCTIONS WITH ANGULAR DENSITY

[CHAP. II

If p(r) = const, then one may choose for 91' the empty set, and for the function 9'(z) the exponential function exp c,z' (C, = T~-ip8I). In the general case, for an arbitrary p*(r) the function 9'(z) must (because of Theorem 2) satisfy in the complement of some CO-set the asymptotic equation

lnl9'(rei')1 ~ "",COS p( (J - (J,)r>. Hence, the case when p is an integer has been reduced to the construction of a set 91' with the indicated properties, or, what is the same thing, of a function !p{z) which is the analog (for an arbitrary proximate order p*(r» of the function exp c,z'. For the construction of the desired function, we note that from the definition of a strong proximate order p*(r) and of the corresponding function L *(r) it follows that r P+ 1 [L·(r»), =r P [1}~(lnr)-O~(lnr)]

,a_(II1",-&,
If we set (i = 1,2),

we obtain rP+l

where

= X2 (r) 1.,.<') = O. r P (,.)

[L· (r»)' lim

,.~QO

Xl (r).

(2.53)

We shall also show that the functions li(r) can be made monotone increasing by subjecting them to an unessential transformation. Indeed, from the definition of li(r) it follows that

1.~ (r) = rP-1L* (r) O~ (In r) [p + 8~ (In r) + &~ (In r) - &~ (In r)] • &i (In r)

The quantity within the .brackets tends to p as r --+- 00, and hence the derivative X;(r) will be positive for sufficiently large values of r. The function l~r) will therefore be monotone increasing after some value of r. For the construction of the set 91' we mark first those points bkl on the ray arg z = 0 at which the function Xt(l') takes on integral values, and on the ray arg z = wI p those points bu at which the function XI(r) takes on integral values. Let us denote by nl (r) (j= 1,2) the number of the points bit; whose moduli do not exceed r. Then it is obvious that nj (r)

= [Y.J (r»)

(j= 1. 2)

and the angular density of the constructed set is zero. On the other hand,

~ bil+ ~ biz' I "k1 1<"

I"bl<"

,.

= ~ Ib,,11-'- ~ Iblr2 I-'= I "lt11 <,.

1"_1<"

f dn~t). 0

Sec. 4]

121

CONSTRUCTION OF AN ENTIRE FUNCTION

where net) = n2(1) - nI(t). It is obvious that net) = t P+ I[L *(1)]' + 0(1) and net) = 0 for some E > 0 and 0 < t < E. Integration by parts yields r

_1_ pL*(r)

S

dn(t)

tP

=

r

nCr)

pr P(r)

r

+_1_ {SIL*(t»)'dt+ S ,(t) dt}, L *(r) tp+l

o

• I (2.54) 1 where !pet) = net) - t P + [L *(1»)' = 0(1). Let us include in the set of the points bl'; the point b defined by the equation co

lrP = L*(a)-

S:p~!dt . •

Having constructed 15(r) for the corresponding canonical product, we obtain l5(r) = I + 0(1). By multiplying each of the numbers bltl and b by Tjl/p ei6r, we obtain a set {bit} of zero density such that

· {I I1m L*(r) r-+co

P

~

~ Iblcl
b-P} Ie = 'e,e -ip8,.

We have thus constructed a set 91' whose canonical function has completely regular growth and has the required trigonometric indicator. This completes the proof of the theorem. We note that the function constructed with the given indicator has completely regular growth. REMARK. If the given trigonometrically convex function is non-negative,' then in the assertion of Theorem 3 one may replace, in the case also of integral order, the strong proximate order by an arbitrary proximate order. In order to show this, we construct for the function L-l(r) a majorant L!(r) (the possibility of this construction was shown in § 12, Chapter I), and we set L *(r) = [L!(r)]-I. Then we find that

-.- Le(r) 11m -L() =1.

r-+ co

r

After this we construct the entire function F(z) of strong proximate order p*(r) (r PO(,) = rPL *(r» with the given indicator h(O) and of completely regular growth. Let E be the exceptional EO-set for this function. Then we obtain i8 e I-mj In I F{re i8) I -_ 1·1m In IF (re • ) I 1-' 1m L (r) -_ h (") u. P P r-+co r L (r) r-+<.o r L (r) r-+co L (r) rEE

rEE

Next, we can assert on the basis of a theorem of Bernstein (see § 18, Chapter I) that h,(O) = h(O). • It may be that the requirement of non-negativeness is superfluous.

122

ENTIRE FUNCTIONS WITH ANGULAR DENSITY

[CHAP. II

S. Asymptotic Representation of Entire Functions with Regular Distribution of Zeros (Proof of Theorem 4) I n order to obtain the asymptotic representation (2.08) of the function j(rei~ with a regular distribution of zeros, one must investigate the function argf(rei~. We shall restrict ourselves to the case when p is not an integer, because the integral case can be investigated in an analogous manner. 9 Our investigation will again be based, as in § 2, on Theorem 25 of Chapter I, but in order to complete the argument we shall need in this case the following extension of Lemma 1. 10 LEMMA

6.

If the set {at} h.as density

and the functions f(z) andf~(z) are defined as in Lemma I, then for every e > 0 and for any z = re iO helonging to some closed set of ordinary raysll one can find a () > 0 such that

<

I arg I (z) -

Here

15

depends onZv

PROOF.

011

arg 10 (z) I ErP (r). e and on the set of rays 9Jl.

(2.55)

Representing the canonical product 00

in the form of the product of the three functions

I.(z)=

II a (:k ; p),

/ akl<'"

J

(z) =

,I,,(z) =

IT 0 (1- ; p), " II 0(:; p)

I ale I >,..

"r
Ie

< (] < I < T < 00, Izi = r) we can write the inequality I arg I(z) - arg I~ (z) I -< I arg I. (z) - arg I! (z) I + Iarg, I (z) - arg, 1° (z) I+ Iarg, I. (z) - arg,/! (z) I.

(2.56)

• Below we shall indicate the necessary changes in the argument in the case of integral order. )0 When p is an integer one needs an analogous extension of Lemma 4. 11 For the definition see § 1.

Sec. 5]

E,

123

ASYMPTOTIC REPRESENTATION OF ENTIRE FUNCTIONS

From Lemmas 7 and 8 of § 7, Chapter I, it follows that for any given positive for a sufficiently small u > 0, and for a sufficiently large T > I,

Iarg/" (z)- arg/~(z) 1< ;

rp(r)'j

(2.57)

largJ(z)-arg./O(z)I<; rP·r). For an estimate of the third summand in (2.56), we note that expression

In

the

(u + ;2 + ... + ~)

argQ(u; p) = arg(l-u)+arg

the second summand is a uniformly continuous iunction oi u when liT ~ lui ~ l/u while the first term is a uniformly continuous function in the cut ring (I/T <: lui <: I/o, <: arg u < 2'17). In crossing over the cut (1, +00) along the positive axis, this function increases by 2'17. Let us express J.,(z) as the product of the two functions

°

d",s(z) =

II

Q (: ;

"r<1 Gle I <.r I argale-Sl:> 6

II

;;;e(z) =

p).

Ie

Q (: ;

..r<1 aleJ <.r largale- I
p) .

Ie

The existence of the density of the set of zeros implies that for some C n(r)
or that n(tr)

If (II

(J

>0

>0

(r>O)

< C.rp(r).

is sufficiently small, then for it is true that

I'PI

~ (J

and

I'P - 'PI I < (J, and for

~ S ~ T- I

Iarg Q (sei'i'; p) - arg Q (sei'i'.; p) I <

S::•.

and we shall have ~

larg./..,s(z)-argd... s(z)1

< 6C. n(tr) 0 and 18 - 'PI < (J

Iarg 0 (se l "'; p) -

arg a (se"; p) I < 31t.

Therefore,

Iarg.h. s(re fl ) -

arg.if. I (re iS ) I

< 31t(n(tr.

I,+S)-n('tr, IJ-S»).

124

ENTIRE FUNCTIONS WITH ANGULAR DENSITY

If arg z = () is an ordinary ray, then for a given

EI

[CHAP. "

one can find a 15 such that

(2.59)By the finite-covering theorem we can, for a given EI , choose a 151 such that the inequality (2.59) will be satisfied if 15 < bl for all () E 9)1. Having taken ~I and b sufficiently small, we obtain

I arg.;; 8 (re i8 ) -

arg.

7:.

9 (re 18 )!

< i- r P

(~ E!In)

(t)

and, combining this with (2.58), we obtain an estimate for the third summand in (2.56)

!arg. /0 (re''9) -

~'9

arg. /0 (re' ) I

< 3" r P

£'r)

for () E 9)1 and b sufficiently small. This inequality in conjunction with (2.57) yields

Iarg 1(re iB ) -

arg 18 (re i9 ) I

< erP

(r)

for () E,9)1 and b sufficiently small. Lemma 6 has ~hus been established. Let us now proceed with the proof of the theorem. From Theorem 25 of Chapter I it follows that if the zeros of the function j(z) are located on a finite number of rays arg z = 'P; with densities !:l.; (j = 0, I, ... ,n), then for I() - 'P;I ~ 'YJ > the following asymptotic relation will hold uniformly:

°

n

arg l(re i8 ) ~ -J_1t- ~,:1, sin p (6 n7tp ~

S

J

~'i -It).

(2.60)

i~O

In this case of regular distribution of the zeros we choose for a given positive a positive b so small that, for I'PHI - 'P;I < b (j = 0, I, ... , n) and 'Pn+1 = 'Po + 217, the sum on the right hand side of this equation will differ from the corresponding Stieltjes integral by less than E/2. Furthermore, if b is so small that for () E 9)1 E

I arg 1(re4~) -

arg f (re la) I

< ; rp(r).

then for () E 9)1 and () not belonging to the interval I'P; - ()I for some r.,,9R and r > r.,,911 the inequality

I

arg I(rei~) -

f

8

1trP(r) Sin 1tp

sin p (6 -.} -It) d~ ('f)

< 'YJ we shall have

I<

uP (r).

8-21<

Selecting another subdivision of the interval (0, 217) by means of the points

'Pi (j = 0, I, ... ,n) such that 1'P;+1 - 'Pjl < 15, and such that the intervals I() - 'Pjl < 7J have no points in common with the intervals I() - 'Pjl < 'YJ, we find that the displayed inequality holds for all () E 9)1 and all r exceeding some number reIJI.

Sec. 6]

ENTIRE FUNCTIONS WITH REGULAR SETS OF ZEROS

125

From this and from Theorem 1 there follows directly the truth of Theorem 4 when p is not an integer. The case when p is an integer differs from the above only in that in place of the functionf(z) one must consider the function/,(z), and in place of Theorem 25, Chapter I, one has to use Lemma 9 of Chapter I. Also, one has to take into account the equation eip. "

.

11m -L() o,(r)='t,e

,.-+00

ip (8-8.1

r.

r

6. Entire Functions with Regular Sets of Zeros (Proof of 1beorem S) The asymptotic formulas established in §§ 2, 3, and 5 are valid outside some exceptional CO-set of circles, which is not effectively constructed. In this section we shall prove that the zeros constitute an R-set, and we shall show that the asymptotic formulas (2.03) and (2.05) are valid outside a well-defined CR-set. The precise formulation is given in Theorem 5 (see § 1). For the proof of Theorem 5 we shall need two lemmas. LEMMA

7.

Let {an} be an infinite set ofpoints having a limit point at infinity,

and let Fa(z)

= II(l -

:J.

where the product includes those factors in which the numbers an satisfy the inequality Ian - zi <; ~r, Izl = r, and 0 < ~ < !. Then for z "" an we have the following inequality, If"

IlnIFI(z)I/
f

(2.61)

n't(t) dt,

o

in which nz(t) is the number of the points an in the circle of radius with center at the point z, and where CIJ is a positilJe constant depending only on ~. PROOF.

From the form of the function F,iz) it follows at once that

I'n!Fa(z)I/
It is easy to see that

~ Inll-~I· ,-G" I .;; If"

I,.

~

In I an - z I = fIn tdn,(t)=n,(8r) In 8r-

IG,,-' I.;; I,.

0

and

Inl

n

I G,,~';;lr

0;-11>-",(or)ln(1+8)r.

If"

f 0

n't(t) dt

126

ENTIRE FUNCTIONS WITH ANGULAR DENSITY

[CHAP. II

From these inequalities it follows that 1)

(ar

nz(t)

IlnIF.,(z)1I < In ( 1 +"6 nz (6r) + Jo -t- dt. LEMMA 8. If the poillt set {an} is an R-sel and if F~(z) is the same as in Lemma 7, thell for erer), pusitire E, for a sufficiently small posit ire number ~, and for allY Z I)'ill~ outside the exceptional C Il"circles and outside some circle 1=1 ~ r., we hare the inequality

lIn I Fa (z) II < ar P (r).

(2.62)

PROOF. Suppose the condition (C) (see § I) is satisfied, and suppose that the point;; does not belong to any of the circles mentioned in that condition. With =as center let us draw a circle K, of arbitrary radius I > O. Since the center of this circle does not lie in any of the exceptional circles. the radii of the exceptional circles with centers inside this circle are less than t. Since no two of the exceptional circles have a common point, the number n.(I) of their centers within the circle K, is less than the ratio of the area of a circle of radius 2t to the area of the smallest one of the exceptional circles with centers in XI' If we denote this smallest radius by I"., then for t < r we have

Lt "

> d(r -

plt+r)

t)(r +t)- - 2 -

and consequently, for t '" ~r and for sufficiently large r, 47ttl!(r+t)P(r H )

n,,(t)<

7td 2(r-t)2

8 (1 +o)Ptl! (1-&)2 r p (r)-2.

< d3

An analogous estimate can be obtained under condition (C') by comparing the diameters of the circles. Substituting the derived estimate for n.(t) into (2.61) we obtain (2.62). This proves the lemma. 12 We shall now proceed with the proof of Theorem 5. The function f(=) satisfies the asymptotic equation (2.03) or (2.05). Here the inequality In I !(re ie ) (H (6) +eJ rP(r) (0 6 2'1t) will be satisfied for all r greater than a number r;, and the inequality

1<

< <

In l/(re i8 ) I > [H(6)- eJ rP (r) will hold in the complement of some CO-set. II The conditions (C) and (C') are required only to guarantee the validity of inequality (2.62), which is obtained from the estimate of n.(t) in view of Lemma 7. The conditions (C) and (C') can therefore be replaced in Theorem S by any other conditions that will yield a corresponding estimate for n.(t). In particular, when p(r) = I, one may, for example, require in place of condition (C)that for some d > 0 and for arbitrarynandk the inequality la••• - a.1 > dk be satisfied. Then n.{t) < (Ild)t and if Iz - a ..1 > d. (d. > d, n = 1,2, ... ) the inequality (2.62) will hold.

Sec. 6]

ENTIRE FUNCTIONS WITH REGULAR SETS OF ZEROS

127

We need to show that the second inequality is valid everywhere outside the CR-circles when r > r;. For this purpose we pick a z outside the CR-circles such that Izl > 2r;, and we define the function Fa(z,u)=

II (l_zt

I an - 81 " ~r

U ).

n

It is obvious that F~(z, 0) = F~(z), and that the inequality In IFiz, u)1 < 0 will hold if ~ < t and lui < ~r. This implies that within this circle but outside the exceptional CO-set the function

<Pa (z, u) =

/(z+u,9l) Fa (z, u)

satisfies the inequality In I<Pa (z, u) I

> (H~ ('J.) -

e) Iz+ u IP (1-+"1),

in which 1p = arg (z + u). From the continuity of Hm(1p) it follows that for a sufficiently small positive ~ we have In I<Pa (z. u) I > (H!R (6) -

(2.63)

2e] rP (r)

in the same region. We shall now show that for sufficiently large values of r this inequality is valid when u = 0 even if z belongs to the CO-set. Indeed, for a given ~ > 0, and a sufficiently large r, the sum of the diameters of the circles of the CO-set which lie in the circle lui < ~r is less than ~r, and hence there exists a circle lui = 7' (0 < 7' < ~r) that does not intersect the CO-set. The inequality (2.63) is satisfied on the circumference lui = 7', but since the function rp~(z, u) has no zeros in the circle lui <; 7', it must also be satisfied at the center. Thus In I/(z, 9l)1 I Fa(z) I

> (H!R (6)-e]rp(r)

and in view of the inequality (2.62) In

If(z,

!R) I> (H91(6)-3s] rP(r).

This inequality, which we have proved for all sufficiently large rand z not lying in the CR-circles, taken in conjunction with the asymptotic inequality (z

= reil )

(see Chapter I, Theorem 28) establishes the first assertion of Theorem 5. The second assertion concerning the argument of fez, 91) does not require any additional argument, since the asymptotic equation for argf(rei ', 91) is not connected with any exceptional sets of circles but is valid on any closed set of ordinary rays for all sufficiently large r.

128 EXAMPLE.

[CHAP. II

ENTIRE FUNCTIONS WITH ANGULAR DENSITY

Let Om ...

= m + in (m, n = 0, ± I, ±2 •... ) •

tt

Z) e-2- + - a(z)=zII,(1 -,-)III,n

222

m,lI

..... n

(the prime indicating that the values m = n = 0 are excluded). The set of zeros of this function has angular density ~

III (.}) =..!...

2

with the index p = 2. In view of the continuity of the function .1(1p) there are no exceptional rays. The condition (B) is satisfied because d,,(r) = O. Furthermore, the set of the zeros is an R-set because the condition (C) is satisfied when d < 1. Thus, outside the circles with centers at the integral points Om.n. and with arbitrarily small radii, we have, in view of the formulas (2.08) and (2.10), the asymptotic equation

lnla(reiB)1 ~ ;

,2

which is satisfied uniformly relative to 0 in the interval (0

~

0

~

7T).

7. Theorems on Equicontinuity (Proof of Theorems 6 and 7) First of all we point out the close connection that exists between Theorem 6 and Lemma I proved above. In Lemma I we showed that if the set of zeros of an entire function fez) of proximate order per) has a density. then under a sufficiently small change of the arguments of the zeros 0/c. the function In If(rei~1 will undergo only a small change asymptotically. Since the original factors of the canonical product depend on the quantities

3....- = _r_ ei \8-<\>k) ak

(6

= arg Z.

lakl

Yk = arg ak' k

=

1. 2. 3 •... ).

a small change in 0 is equivalent to a small change of the arguments of the zeros O/c (k = 1.2•... ). Thus. if the set of zeros of the function fez) has a density. then to every positive E there correspond a lJ > 0 and an '. > 0 such that if 102 - 011 < lJ. , > '" and ,ei81 , rife, do not belong to some CO-set (or if r does not belong to an EO-set that is independent of e). then

I h"

r

(6 1)

-

h"

r

(6 2)

I < I.

For r <: r. and outside the EO-set. the function hl.r(O) is uniformly continuous in , and 0, and hence for some ~2 > 0 and 101 - 021 < lJ2 the same inequality will hold. It is sufficient to take 6 = min (61, lJz). Thus if the set {a/c} of the

Sec. 7]

129

THEOREMS OF EQUICONTINUITY

zeros of an entire function of proximate order p(r) has a density. the assertion of Theorem 6 follows from Lemma I, as has just been shown. 13 In the proof of Lemma I we made use of the density of the set of zeros of the function only in the estimate of the quantity lIn IR,Il(z)lI. For the proof of Theorem 6 without auxiliary hypotheses on the set of zeros, it is therefore natural to make use of the general estimate for lIn "1<1> R,II(Z) 1 given in Lemma 2. This estimate involves the quantities n", and n",H' namely. the number of zeros of j(z) in certain rings Km: Rm{l - (J) < Izl <;; R",(l + (J), and K"'+1: R,n+! (I - (J) < Izl < R m+1(1 + (J). If the set of zeros of the function j(z) does not have a density, then the roots can be strongly concentrated in certain rings; however, if one notes (see Lemma 4, Chapter I) that for some constant c

I

n (T)

< CTWI,

(2.64)

then it is easy to understand that the relative measure of such rings can not be great. Let us give an exact evaluation of the -relative measure of such special rings. We, shall denote by Iq the intervals

< R ~ Rq{1 +~) G+ :)', 0 < ~ < i)

Rq(I-~)

( Rq =

and by Kq the rings Rq (I -~)

< I z I ~ Rq (I + ~).

Denoting, as before, by nq the number of points of the sequence {all} in the ring Ktl , we shall say that an interval III' or a ring Ktl , is a singular interval, or a singular ring, if

(2.65) where !l.q denotes the length of the interval III' Let n(t) denote the number of points of the sequence {aft} in the circle Izl " t. We shall prove that the upper relative measure of the set of singular intervals will tend to zero when (J -+ O. For this purpose we consider the sum SB

= R-p(B)

~'n , B

qj

qj

where the summation extends over those values of the index q which correspond to the singular intervals. 11 In this case we have obtained the assertion of Theorem 6 in a stronger form, since in plaec of the execptional set E of arbitrarily small upper relative measure, we have used an E--set.

130

ENTIRE FUNCTIONS WITH ANGULAR DENSITY

[CHAP. II

From (2.65) we find that 1

S

. . . . ~-z

B ,..

~!!.

~

p(Bt} I Rq j j

1

_1 ~ ~- 2R- 1

~

qj RP,B) RIJ.J ~

~

p(B q ) Rt} j

!!.

qj

~j......,.... RPIB)

!!2
R.t
From the monotone character of the function r P(7) it follows that if Rfll then

> R/2,

In this manner we obtain the inequality 1

~

< 2P+2~ZSB ~ •

!!.q.

B

(2.66)

J

z
< c,

Returning now to (2.64), we show that SIt (2.66) implies that

1

~

!!.

~

B

and that the inequality

q. J

< 2P+2 cQ2 R2' ~

2
Substituting in this equation for R first R/2, then R/4, and so on, and adding the resulting inequalities, we find that for some Ro > 0

~

!!.

< 2P+2C~}R

Bu
and finally B

~
IJ.j

We have thus shown that the upper relative measure of the set of singular intervals tends to zero when {J - O. Let us now proceed with the estimate of the function lin ICI>R./I(Z)!I outside certain exceptional rings. We pick out from the sequence of the intervals II all singular intervals and all intervals adjacent to singular intervals and denote this set of intervals by E'. Then we shall have for 0 < (J < 1 the following inequality: 1

m-(E')

< 2P+5C~2.

(2.67)

Sec. 7]

Rm

131

THEOREMS OF EQUICONTINUITY

Let us return to Lemma 2. If the number R does not belong to E' and if < R < Rm+l' then the intervals 1m and Im+l are not singular intervals, i.e.,

Hence, if z is such that Ilzl - RI < (3R and does not belong to the set C of exceptional circles, then it follows from Lemma 2 that

Ilnlq,R.~(z)11 < 2:1{(R~RtII\ + R~l:~+t\)(2In ~ + where Bp

Bp),

= 4 + 3Pp, which implies that 1

lin I R. ~ (z) II < 22P+2~2( 2 In t+ Bp) rP(rl.

(2.68)

Here the upper relative measure of the set E" of those values of r for which the circumference Izl = r does not intersect the exceptional circles of the set C will not exceed the quantity {3(l + (3)3(l - (3)-2. Let the numbers fJ > 0 and e > 0 be given. We choose {3 so small that

2PHC'3~ <1 1-, ~q+~)S(I_~)-2<1-' 22P+2~2 (2 In!+ Bp ) fl If we now set E

=

E'

I

(2.69)

< -=-10'

+ En, then m!(E)

<-r"

and if Izl E E,

lin IcfR.~(z)11

< ;or Hrl .

(2.70) Let us now express the functionj(z) of the proximate order p(r) in the form of a product of five functions as was done in subsection I of the proof of Lemma 1. We shall then have

lin I1(re iO .) I-In I1(re iOI) \ \ < /In 110 (re iO.) I-In 110 (re iO.) II lin IJ(re io.) I-In l·rI(re io.) II lin II~ (re iO.) 1- In Ilo~ (re iO.) II lin 1~~/(reiO.) I-In 1~~/(reiOI) 1/ lin I ~ (re iO.) I-In I ~(rei8.) 1/

+

+

+

+

and we must estimate the value of each summand of the right hand side. From (2.70) it follows that if r E E,

Ilnl~(rei9')I-lnllI(rei8')II< ;rp(r).

(2.71)

132

ENTIRE fUNCTIONS WITH ANGULAR DENSITY

[CHAP. II

Furthermore. from the inequality (2.20) we have the asymptotic relations

If. (re ie ,) I-In I f. (re ie") II < i

rP (r),

lin !. f(reie,) I-In l.t(re io,) 11< ~

rP(r).

lin

)

(2.72)

Finally. from the uniform continuity of the functions of 0 In 10 (se i9 ; p) I in the regions (1 - ~)-1 <. S 0- 1, ~)-1 .. -I<: s <: (l

+

<

and from the inequalities (2.64) it follows that, for a sufficiently small 15 and for 101 - 021 < 15,

Ilnlf.~(,ei~')I-lnlf.~(reio!)II<; lIn l.~ f (re iQ .) I-In I,~ f(re i9,) 11<

101

>0

r Hr1 , )

ir

(2.73) Plr1 •

From the inequalities (2.71), (2.72) and (2.73) it follows that. for, E E and 02 1 < 15,

-

(2.74) We have thus proved that for every pair of positive numbers "I and £ there corresponds a set E with an upper relative measure less than "I and a number 15 > 0 such that if r E E and lOt - 021 < d. the inequality (2.74) will be satisfied. We must still prove that if one omits a certain fixed set E of values of r with an arbitrarily small upper relative measure, then for every £ > 0 there exists a tJ > 0 such that if lOt - 02 1 < 15, then (2.74) will hold. For the construction of such a set E we choose for the given "I two sequences of positive numbers: £1' £2' • . . • converging to zero, and "It. "12' .•• such that 00

To each pair of numbers CIe , "lie there corresponds a set Ek with upper relative measure less than "lie' and a positive number tJ(t) such that if r E Ele and 181 - 02 1 < 15(1), then lIn I f(re iO .) I-In I f(re i9.) II

< skrP

(rl.

(2.75)

We improve the sets E" somewhat by dropping from each Ele the part belonging to a certain interval (0, rle)' and we do this in such a way that the remaining set E" satisfies the condition mes (E: I

< 2'1"",r

(r

> 0).

Sec. 7]

133

THEOREMS OF EQUICONTINUITY

Then it is obvious that the upper relative measure of the set

E;

will not exceed 'YJ. The set and the number bill are selected in such a way that if, > 'k' r E E. and 102 - 011 < b~1) the inequality (2.75) will hold. On the other hand. if Izl < rIc. and 1=1 E £1' the function In 1/(z)1 is uniformly continuous. and hence the inequality (2.75) will be satisfied for some ,}~2). 19) - 921< 2>, r " rk , and r E E. If we choose 6k = min { BP>, BP>}, we find that (2.75) is satisfied if r E E and 191 - 921< BIe... This implies the equicontinuityof the family of functions "",(9) in 9 if r E E. The theorem has thus been established. Let us now proceed with the proof of the more general Theorem 7 on the equicontinuity of the family h F.r(O) of functions which are holomorphic inside some angle 0( ~ arg z ~ p. We shall first prove three lemmas.

61

LEM~A

9.

Let $(z) be a holomorphic function in the circle

Izl

~

R, and let

max I
M~(R) =

18 1
Let us also suppose that at some point

=0 (Izol

I
~

IR, I

< I)

1.

Then the followillg inequality will be true for the number n~(tR) of zeros of the junction
n~(tR) <. v(t,

t) In

M~(R)

(v (t, t) = In c/~'n -1).

LEMMA 10. If under the conditions of Lemma 9 the function $(z) has no zeros ill the circle Izl ~ R. then the minimum mq,,(tR) 0/ the function 1$(z)1 in the circle 1=1 ~ t R satisfies the inequality

In m~(tR)~ -

vet) In

M~(R)

( .. (t) = 1 11 tl' t1 = In:

t:, ).

These lemmas differ from Lemma 4, § 5, Chapter I, and from the corollary of Theorem 8 in § 6, Chapter I. in that the requirement 1$(0)1 = I has been replaced by 1$(zo)1 = I, and IZol <; IR. Both these lemmas reduce to the propositions of Chapter I if the circle Izl <; R is transformed into the unit circle and the point Zo is replaced by the origin. We shall prove only Lemma 9. Lemma 10 can be proved in an analogous way. PROOF.

The function

z=

R'If.

+ Rzo

R+c.o

134

ENTIRE FUNCTIONS WITH ANGULAR DENSITY

[CHAP. II

transforms the unit circle I~I ..;;; 1 of the ~-plane into the circle Izl .s;; R. Here the point ~ = 0 will go into the point ZOo Substituting the function

~ (~) =
J'7n ('t) d, =

f

..,

2..

1 2~

o

In 19 (e") 1d,J ..;; In M~(R)

0

and because of the monotone character of the function n",(T) for 0 have 1 n'f'(tl)In

< tl < 1, we

It < InM~(R).

When tl = (t + 1)/(1 + tl) the image of the circle I~I ~ tl contains the circle Izl < tR. Therefore. n(Rt) .s;; n",(t1) and hence n~(tR)

< '1(/.

t) In M",(R).

Lemma 9 makes it possible for us to find an estimate of the number of zeros within a sector of a function satisfying the conditions of Theorem 7. Namely, the following lemma is true, LEMMA II. Suppose that F(z) is a holomorphic function of proximate order p(r) inside the angle ex < arg z < fJ and suppose that for some positive number N, and for some positive number I < 1, and for all sufficiently large values of r, each circle of the type .,,+~

Iz-r/-2-1
ZI

=

r1ei61 in which InIF(Zl)1 >_NrP(rl,

Then for ~ > 0 the number n(r, ex in the sector ex + ~ < arg z < {J inequality

+ ~, {J ~,

~)

of the zeros of this function

Izl .s;; r, will satisfy the asymptotic

in which h=

. max

1h,l<,(IJ) 1

,8,~

and

K(~,

iX,

{J. I) is a constant which depends only on the indicated quantities.

PROOF. We make the following preliminary remarks. The assertion of the lemma applies to an arbitrary angle inside the angle (ex, {J). Therefore, without loss of generality we may assume that {J - ex is not a multiple of 1T/p. By means of the transformation z = eifPu A one can transform the angle (ex, {J)

Sec. 7]

THEOREMS OF EQUICONTINUITY

135

into the right half-plane, whereby the function F(z) wi1\ go over into a function of a non integral order in the right half-plane. Here one can select the positive number 11 0 in the circle Iz-RI
Hence we may assume, without loss of generality, that IX = -71'/2, {J = 71'/2, and p is not an integer, and that for al1 sufficiently large values of r the circle

Iz-rl
zl.r

=

In

at which

rleiO,

IF (Zl, r) I > -

Nr~ (r,l

>-

N(2rt<2r).

(2.76)

On the other hand, by Theorem 28, Chapter I, the func:ion F(z) must satisfy the inequality In I F (z) I

< (h + e)(2r)P(2r)

(8) (')

(2.77)

everywhere in the circle Iz - rl ~ Ir. From the inequalities (2.76) and (2.77) it follows that for al1 sufficiently large values of r the function , ()_ F(r+z) IJir z t"(z 1. r )

wil1 satisfy, in the circle Izi In I ~r(z)

~

r, the inequality

1< (h +N +e)(2r)P(2r) < (h + 2N)(2r),(2r) (0<8 < N).

Also, for Zo

= zl.r -

r we have the equation 1JI,,(zo)

= 1.

By Lemma 9, the number n.,(kr) of zeros "f the function 'Pr(z) in the circle

Izi

< kr (/ < k < I) satisfies the inequality (2.18)

in which "1(1. k) is a constant depending on I and k. With the aid of (2.78) we must now give an estimate of the number of zeros of the function F(z) inside the sector Izi " R, and larg zl " (1 < 71'/2. For this purpose we choose a number k (sin (1 < k < I) and draw a circle of radius kr with center at r. This clide will intersect the sides of the anglf! arg z = ± (1 at the points

Izl = r(cola=!::Yk~-sinlla).

136

ENTIRE FUNCTIONS WITH ANGULAR DENSITY

Let us set R the region

=

= Vk 2 -

'cos a and s

[CHAP. II

sin 2 alcos a, and let us denote by Kif.

R(I-s)
of (2.78) we obtain

+

< 'V! (I.

k)(h 2N) rP I"!, where n(KR ) is the number of zeros of the function F(z) in the region K R , while ".(1, k) is a constant. Making use of the equation R = 'cos a we can rewrite the last inequality in the form n (KR)

n (KR)

<

'1 2

(I. k. a)(h

+ 2N) R (RI,

(2.79)

P

where "2(1, k, a) is a constant depending on the indicated quantities. For given R > 0, we take a sequence of numbers

Ro=R(I+s)-t, R1 =Roq. ...• Rp=RoqP . ... (q=!+:). From the definition of proximate order one

~an

(2.80). easily see14 that for some

'0 and '2 > '. > '0 P

rP Ir.) 1

< (0.)2 rP '2 2

(r.' •

The sequence (2.80) will be assumed to be finite and to satisfy the condition Rp> '0. Writing down the inequality (2.79) for each one of the regions KR and adding these inequalities we obtain p

PP

n(R. -a. a)
+ 2N) RgIRo',

or asymptotically (2.81) where K(I, a) is a constant depending on the indicated quantities. This completes the proof of Lemma II. We now proceed with the proof of Theorem 7. Let an be the zeros of the function F(z) within the angle larg zl (71 (ct < ct. < 7T/2). We assume that these roots have been arranged in the order of increasing magnitude. The canonical product

<

/(z)

=

Ii: a (:,.; p)

(p

=

[pI)

1

is an entire function of no higher than normal type for the proximate order p(r). By Theorem 6, to every positive number TJ there corresponds a set £'1 '0 Indeed, [In rP('}]' = p(r) r

+ p'(r) In r

=

er + o(!) > .! . Integrating both sides of this r 2r

inequality from r, to r .. we obtain the requ;-ed result.

Sec. 7]

THEOREMS OF EQUICONTINUITY

137

< TJ] such that if r E E~ the function hl.r(O) will constitute an equicontinuous family. The fraction

[m·(E~)

F (z)

X. (z) = f

(z)

is a non-null holomorphic function of type no higher than normal and of proximate order per) inside the angle larg zl ..;; 0'2'0' < 0'2 < 0'1 (Chapter I, § 8). Therefore for some N(~) > 0 we have the asymptotic inequality

In Ix. (z) I < N~l)rP (f).

(2.82)

On the other hand, from the condition (2.12) of the theorem, and from the upper bound of In /f(z)/ In I fez) 1< a) r P (,.).

(0,+

it follows that each circle of the type

Iz-rl
In IX. (Zl. ,.) I > N~lrP ("1 (N(;) is a constant).

(2.83)

The function x(z) satisfies all the conditions of the theorem and does not have any zeros within the angle larg zl ..;; 0'2 ..;; 17/2. It remains only to prove the theorem for this particular case. We shall give an estimate of the function lin Ix(z)11 inside the smaller angle larg zl ..;; 0'3 (0' < 0'3 < 0'2)' For this we make use of Lemma 10. From (2.82) and (2.83) it follows that the function 6 (u) = X. (r

+ u)

1.. (Zl. ,.)

satisfies, within the circle lui ..;; r sin

0'2'

the inequality

In 16 (u) I < N~lrP (I")

(N(:) is a constant)

and at the point Uo = z1,r - r of this circle, O(uo) = 1. Moreover, the function O(u) has no zeros in this circle. From Lemma 10 we see that in the smaller circle lui < , sin 0'3 the following inequality is valid:

lIn 16 (u)

II < ~:)rP

(I')

(N~4) is a constant).

Finally, returning to the function x(z), we find that everywhere within the angle larg zl ..;; 0'3 the inequality

lIn I X. (z)

II < Nxr

P (I')

(Nz is a constant) (2.84) is valid. With the aid of the bound (2.84) we establish the equicontinuity of the family of functions of 0 when r > '0 and 'I is arbitrary.

138

ENTIRE FUNCTIONS WITH ANGULAR DENSITY

[CHAP. II

For this purpose we express the harmonic functions In Ix(r circle lui" R (R = r sin 0'2) by means of the Poisson integral:

+

u)1 in the

2"

I

In X(r

+ u) I= ;7t f In / X (r + Re'';) IRe {:::: ~: }dljl. o

After we have chosen k = (sin 0'3)/sin 0'2 and the numbers lui" kR, we have In / X(r

+u

/-In / X (r

l)

Ul

and

U2

in the circle

+ u~ /

and in view of (2.84) we obtain lIn I 1.. (r

+u

l)

I-In Ix (r

+ u<J) II

2K

(2rt

(2,.,

I

l.. fiRe {e'''' + UtR - 1 _

2K

e'' ' -u R-

0

1

1

From this inequality it follcws that for the given e and for lUI - u21 R-I < ~l' lIn I X (r

>

By choosing IS

181 1"

0'3'

182 1"

°

+u

l)

I-In Ix (r

e'''' + ~ -1 }

ei'" -IltR- 1

I

d.I • jI

> 0, and for some lSI > 0,

+ u II < ar (r, • i)

P

(2.85)

sufficiently small, we deduce directly from (2.85) that if 0"3' and 181 - 821< IJ, then

/In I X(re"') I-In Ix (relB.)//

< arP

(f")

(r> 0).

(2.86)

The function F(z) can be expressed in the form F(z) = X(z)/(z).

By Theorem 6, the family of functions In I/(re") I

rI'''' is equicontinuous for r not belonging to some set of arbitrarily small relative measure. Combining this with (2.86), we see that for any TJ > 0 the family of functions h

P.,. (6) = In IF(re rP,r)

il ) I

is equicontinuous in 8 if 181 " 0'3 < TT/2, and r E E'I' where E'I is some set of positive numbers with an upper relative measure less than TJ. This completes the proof of the theorem.

CHAPTER

III

FUNCTIONS OF COMPLETELY REGULAR GROWTH In this chapter we shall investigate a problem which can be considered as the inverse problem to that of Chapter II. There we established that regularity of the distribution of the zeros of an entire function implies regularity of growth. In this chapter we shall show that regularity of growth implies regularity of the distribution of the zeros of the function. l A function F(z) that is holomorphic and of proximate order per) within some angle (IJ l , IJ 2 ) will be called a function of completely regular growth on the ray arg z = IJ if the limit i8

hF (6) = lim In IF (re ) 1 r~oo

rP

(r)

exists under the condition that r goes to infinity by taking on all positive values except possibly those of a set of zero relative measure (an EO-set). We shall sometimes write lim· to express this situation. We shall say that a function F(z) is of completely regular growth on some set of rays R9Jl orn is the set of values of IJ) if the function h

'8

F. r

(6) = In I F(re t )1 rP(r)

converges uniformly to hF(f) for f) E WI when r goes to infinity by taking on all positive values except possibly for a set E9Jl of zero relative measure, this set being the same for all rays R9Jl. The set E9Jl will be called the exceptional set for the given function. We shall say that F(z) is a function of completely regular growth within the angle (0 1, ( 2) if this is true for every closed interior angle, and we say simply that F(z) is of completely regular growth if it is an entire function and is of regular growth in the entire plane, i.e., for 0 <;; f) <,; 21f. The functions with regularly distributed zeros considered in the preceding chapter are functions of completely regular growth. Indeed, it follows from Theorems I and 2 of Chapter II that the function h,.•(f) for these functions converges uniformly to htCf) if r goes to infinity avoiding those values for which the circle Izl = r cuts a circle of some CO-set. It is obvious that this exceptional set of values of r is an £
The basic results of the first three sections of this chapter are due to A. Pfluger [I). 139

140

FUNCTIONS OF COMPLETELY REGULAR GROWTH

[CHAP. 11/

In this chapter we shall show that if F(z) is a function of completely regular growth within the angle (0 1, 02)' then the set of its zeros possesses an angular density inside this angle (for an exact formulation see Theorem 2). Furthermore, if fez) is an entire function of completely regular growth. then its zeros are regularly distributed. Thus, the regular distribution of zeros is a ch;!r::.;.teristic property of entire functions of completely regular growth. If we combine this with Theorems I and 2 of Chapter II, we can conclude that an entire function of completely regular growth satisfies, in the complement of some CO-set, the asymptotic equation. 2 In If (re ts ) I ~ h, (0) r P (r). In order to obtain these results we shall generalize the formula of Jensen (Chapter I, § 5) by considering a holomorphic function in some sector instead of in a circle. If the sector angle is 217, this formula takes on the usual form of Jensen. In § 5 we shall investigate holomorphic functions rJf some proximate order per) with a sinusoidal indicator inside some angle. Functions of this type occur in various applications. The results presented in this section will be used in Chapter VI in the investigation of the zeros of exponential sums.

1. Set of Rays of Completely Regular Growth In the definition of completely regular growth on some ray. an important role is played by a certain exceptional set Eo of zero relative measure which depends, in general, on the selected ray. We shall prove, however, that if the holomorphic function F(z) of proximate order per) within a certain angle has regular growth on each ray of a certain set R<m then one can construct a set E':1Jl of zero relative measure in such a way that for r EE':1Jl' () E IDl, and r ->- 00, In IF (re iS ) I ~ h,1 (6) r P (r) •

We shall also prove that F(z) is a function of completely regular growth on every ray which is a limiting ray of the set of rays along which F(z) is of completely regular growth. THEOREM I. If a holomorphic function of proximate order pCr) within an angle (IX. (J) is a functil'n of complete~v regular growth on each ray of some set R<m belonging to the illlerior angle (IX < 0l <:; () <; ()2 < (J), then it is of completely regular growth on the set of rays RiR. I An analogous assertion is true for all functions which are holomorphic and of completely regular growth within some angle at < arg z < p. We shall not prove this more general proposition.

Sec. I]

141

SET OF RAYS OF COMPLETELY REGULAR GROWTH

I. For the proof we note that if a function F(z) is of completely regular growth on some ray arg z = "y lying within the angle (9., 92), then for every e > 0, and for some positive number I > 1, and for aU sufficiently large r, each of the circles

I

contains a point z

=

I .

.',+'1 < rl SID 92 -2 9.

z-re'-2,eiy for which In IF (re lT ) I

> [hll' (1) -

II r P Ir).

Indeed, the ratio of the arc length cut from this circle by the ray arg z = y to r is a constant quantity, while the ratio of the measure ofthe part of the exceptional set lying on this arc to , goes to zero. The conditions of Theorem 7, Chapter II, are thus seen to be satisfied, and therefore for every positive number TJ there exists a set E'1 of positive numbers with upper relative measure less than TJ such that for, EE'1 the functions hF.r(O) constitute an equicontinuous family on the interval ex < 01 <; 0 <; O2 < p. 2. For a given E > 0 we choose (1 > 0 so that for, EE'1 and \0' - (rl < (1 (01 <; Of < 0" <; ( 2) the inequality

I hll'. r (6') -

hll'. r (6") I

<;

(3.00)

is valid. After that, we select from the set Rm a finite system of rays arg z = OJ (j = I, 2, ... , na) so that the a-neighborhoods of the points 0; will cover the entire set 9Jl. Next we choose the number r. so that for, > '. and, E E, the following inequality will hold: ~

I hP.r(6J)-hp(6J) I < ;

(j= 1.2, ...• ncr),

Adding to the set E'1 all values of, less than '. and all the sets E6 (j = 1, 2, ... , na) we obtain a set £'1 such that m*(£'1) < TJ and I

I hll',r(6)-hp(6) I < 8 (3.01) if 0 E 9Jl, and , E £'1' 3. We now select two sequences of positive numbers, En -+ 0 and TJn -+ 0 (n = 1,2,3, ... ), and determine the sequence of sets En so that m*(En) < TJn and that for, E En it is true that

I hll',r(6) -

h,(6) I

< 8,.

(3.02)

for all 0 E 9Jl. By mathematical induction we construct a sequence of numbers r1 < '2 < ... < 'n < ... (',; -+ (0). We select '1 > 0 arbitrarily. Let us assume that we have chosen the numbers '1' '2' ... , rn-1' Then we pick, n so that' ;> 'fI' and the following inequality is satisfied mes(Ek'"-I)

< ~1J"r,

k <; n - 2,

mes(E':-_.)

< 1J"r,

mes(E':->

< 1J"r

(Er denotes, as before, the intersection of the set E with the interval (0,

(3.03)

r».

142

FUNCTIONS OF COMPLETELY REGULAR GROWTH

[CHAP. III

Such a choice is obviously possible. After the sequence {r n} has been selected we determine a set E such that for r n < r < r n+l' E = En. From (3.03) it follows that for rn < r < rn+ I the inequality mes E' < (2"'n + "'n_l)r will hold and hence that m*(E) = O. Furthermore, it follows from (3.02) that for r > r nand r E E, we have the inequality I hF. r (6) - hF(6) I En (6 E Wt). Thus F(z) is of completely regular growth on WI. In subsection 3 we proved a statement which we shall use below. For this reason we will formulate it as a separate lemma.

<

LEMMA I. If the function F(z) is holomorphic and is of proximate order p(z) within the angle (ex, (1), and if for every pair of numbers E > 0 and 1J > 0 there exists a set E •. ~, whose upper relatire measure is iess than Yj, such that

<

I hF.r(6)-hF(6) I e for r E E.,'1 and ex < () < {3, then F(z) is a function of completely regular growth within the angle (ex, {3). 3 From Theorem I, one can directly obtain the next result. COROLL\RY. The set of rays within the angle ex F(z) is of completely regular growth is closed.

< 01 ~

0 ~ 82

< {3 on which

It follows from this, in particular, that an entire function of completely regular growth on rays that form an everywhere dense set is a function of completely regular growth in the entire plane. 3a

2. A Generalized Formula of Jensen. Investigation of the Function 1;(8) For the determination of the number of zeros of a holomorphic function F(z) in the sector {} ~ arg z ~ 0, Izl ~ r, a number which we shall denote by n(r, {}, 8), we make use of the known principle of the argument. Here we assume that no zeros of F(z) lie on the boundary of the sector. We have the equation

f r

21tn (r. 1If , 6) =

•

o

9

d [arg F (te i8 ») dt dt

+ f d [argr Fdrtt(rei,,») r d~ &

r

_ Sd [arg ~ (tell»)

dt.

o S This lemma remains true if in place of the angle a on a single ray.

< (J O. If p(r) --+ 0, then as Gol'dberg and Ostrovskii [21 showed, the set of rays of completely regular growth of an entire function is either empty or covers the whole plane. It was remarked by GriSin [1) that the second case occurs when In MJr) - o(ln2,-).

Sec. 2]

143

A GENERALIZED FORMULA OF JENSEN

By the Cauchy-Riemann formula we transform the last equation to r

Sd {In I F(tei'f) I}'I'=& dt d

«(J) = _ 2-rrn(r, ,I,

t '¥

o r

I} + fd{In I F(tei'f) td •

'¥

o

Assuming that IF(O)I

=

-~ '1'-

dt+r

.

8 r d. dt(lnIF(le1'1')I}t=rd'fl.

&

I, we introduce the function r

J~(a)= rlnIF(t~)ldt . .'

t

6

This function will play an important role in the present chapter, and we shall consider some of its properties in this section. In the proof of theorems in this chapter we shall frequently assume that 1F(O) 1 = I. This assumption will not reduce the generality of our results. Making use of the function J;(O), we may rewrite the form.Jla for the number of zeros of the function F(z) within a sector in the form 2-rrn (r,

a,

0) =

- { dd

'¥

J;(SO)}

+ {dd'¥ J;(SO)}

"'=&

",=8

8

+ r f {d In I ~?e1'P) I }

t=r d'fl.

&

Dividing each side of this equation by 21Tr, integrating the result from zero to r, and introducing the notation r

N(r. D, 6)= fn(t't&' O)dt, o

we obtain r

N(r, D, 0) =

r

f

2~ [:, J~('fl) d:],.p, -

d1t [:,

o

f JF('fl)~t=&

0 8

+ ~ rIn IF (re") Id'fl.

(3.04)

&

This formula yields a generalization of Jensen's formula, and for 0 = {} + 21T it goes over into the ordinary formula of Jensen. We shall use this formula for proving the existence of the angular density of the set of zeros of a function of completely regular growth, and for obtaining asymptotic formulas for the number of zeros of these functions within a sector.

144

FUNCTIONS OF COMPLETELY REGULAR GROWTH

[CHAP. 11/

For this purpose we first investigate the asymptotic behavior of the function J~(O). LEMMA 2. If F(z) is a holomorphic function of proximate order p(r) within the angle IX " arg z " p, and if the function ;s of completely regular growth on the ray arg z = 0 (IX < 0 < P), then the lim;t

11m ,-p(r)Jp(6) = .!..hp(6)

(3.05)

p

r~Q)

will exist. This assertion can be established quite simply if the function h

P.r

(6) = In 1F e,e") I ,f(r)

has no exceptional set and hence the limit hp(6) 11m hP.r(6)

=

r~Q)

exists in the ordinary sense. Indeed, in this case th~ equation (3.05) is a direct consequence of property (5) of a proximate order. In the general case, when the function has an exceptional set, the proof is considerably more complicated. We note that from the definition of the indicator hF(O) and from property (4) of a proximate order it follows that lim

r~Q)

,-p (r)Jp(6) ~.!..p hp(6).

(3.06)

One needs only to prove that lim ,-p(r)J,(6»! hp(6). r:+Q) P To establish this inequality one has to estimate the function In IF(tei~1 from below. On the complement of the exceptional set E we have the following asymptotic equation:

In IF (te iS) I ::::::l hp (6) tP(f). The problem, therefore, is reduced to the determination of a lower bound on the set E. It is natural to attempt to do this on the basis of the general Theorem 11 of Chapter I on estimation of a holomorphic function from below. First we shall consider the case when 0 = 0 and h~O) = O. From the continuity of the indicator h~O) it is easy to prove, in this case, that for some d > 0 and 101 < 2d we have the inequality Ih~O)1 < £/4 and that, in view of the asymptotic inequality In IF (re") I

< (h p (6) + ~) ,p

(r) ,

we have the following asymptotic inequality within the angle

InlF(reil)l <

; ,,(rl.

101

< d: (3.01)

Sec. 2]

145

A GENERALIZED FORMULA OF JENSEN

Let E. be the subset of the positive rayon which In 1 F (t) I

< - 4"e

(3.08)

tP(l) •

Since the function F(z) is of completely regular growth on this ray, the relative measure of the set E. must be zero. This set can therefore be included in the set l. of intervals l" (k = I, 2. 3, ... ) of zero relative measure. Let us denote by dIe the length of the interval l" and by R" the abscissa of the right end of this interval. With the point R" as center we draw a circle of radius 4ed". From the fact that m*(l.) = 0 it follows that

Urn

r-+oo

(.!.r

~

~

dk ) = O.

Bk
and hence that

· _R dlc = 0 . I1m

k-+ao

(3.09)

Ic

Now we apply Theorem II of Chapter I to the function
=

F(Rk+ u) F (Rk)

in the circle C,,: lui ~ 4ed,.. For all sufficiently large R,., in view of the inequality (3.09) and an inequality opposite to (3.08) at the point R", the function 'P,.(u) satisfies in CIt the inequality In 1
< i (R + 4ed t k

k

(Ric HetJ/c)

+ i Rt

(Rlc l •

Making use of property (2) of a proximate order, we now obtain In I
< aRt (Rlcl •

By Theorem II of Chapter I we see that in the circle lui <; 2d,., but outside the exceptional circles with sum of their radii less than 47Jd,., the following inequality holds: Returning now to the function F(z) we may conclude that in the circle Iz - R.:I <; 2d,., but outside the exceptional circles with sum of their radii less than 47Jd".

(3.10) When 7J < t we find a circle CI: with center at R", and with a radius whose length lies between dIe and 2d,.. This circle must not intersect the exceptional circles. On the entire circumference which includes the segment d,. inequality (3.10) will be satisfied.

146

FUNCTIONS OF COMPLETELY REGULAR GROWTH

[CHAP. III

The circles which are bounded by the circumferences CI: may intersect, and they may form "clusters" of regions np. The diameter /5 p of each region will not exceed the sum of the diameters of the constituent circles. Therefore, if Rp - max. Eo.lzl,

and Op 11m 7["=0.

p-+a>

(3.11)

,

Let a" be the zeros of F(z). We set cpp (z)

=

II (1 - : )

a Ell n

n

p

and

9p (z) = F (z) 4!;1 (z). From (3.11) it follows that if P is greater than some fixed number Po, then

Inl4!p(z)1

<0

(z

E

Up).

Combining this with (3.10) we see that for all sufficiently large values of p we must have on the entire boundary of np the inequality (3.12) The function In 1V',,(z)1 is harmonic in np, and hence, by the minimum principle, the inequality (3.12) must hold in this entire region. Let Ip be the interval which is obtained by cutting the region np with the positive ray, and let E. be the set of all such intervals. It is obvious that m*(E.) - 0, and E. ::::> E,. From (3.12) we see that on each interval Ip In IF (I) I

> In 14!, (t) 1- mat' (')

+ 1].

Furthermore, if tEE.,

where m - 2[H(i)

In W(t) I ~

(t

E

Ip),

- : 1'(".

From these inequalities it follows that for sufficiently large values of r 0

f r

Fo

In I ~(t) I

f r

dt> -me

tP(t,-t dt

r.

(3.13)

Sec. 2]

147

A GENERALIZED FORMULA OF JENSEN

Next, from the definition of the function 11> p(z) we have 1 lrlnl~p(t)1 t dt>- Rp-~p

I

B

~

fp

~

t

I

1 di.

(3.14)

For the estimation of the right hand side we note that if R - <5 <; a

< R, then

I'

In I tile I -

~E~ Bp-~

B B

a

f I~ - I

f

1 dt = a

In

B-3

In I u - 11 du

B-3 a

B-a

f

=a

48(ln2~ -1).

Inlvldv>-2a!o6/2aInvdv;;;'

B-a- 6

a

Substitution into (3.14) yields

rlnl~p(t)1

II

48p

dt

t

>- ~ - 8p

(c5p

In 2~ -

)

1 np'

II

where np is the number of zeros of the function F(z) in the region Op. Returning now to (3.13) we may write r

r

In I F (t) I dt

.

t

o

> _ ~ er~

(r)

P

and therefore

f r

In I ~ (t) I dt

> _ ; erP (r) -

o

(

~

np )

!

+ 0 (r

Rp-r.p<"·

It is obvious that for every

~

(1

Bp-6p <.r

>0 np

<. n(r,

-a, 0)+0(1).

Furthermore, by Lemma 11 of Chapter II, we have n(r, -a, a)

< cyP(r),

p (r».

(3.15)

148

FUNCTIONS OF COMPLETELY REGULAR GROWTH

[CHAP. III

where cF is some constant. Returning to the inequality (3.15), we obtain

,.

f In!~'

(t) I dt

> -(i+ CF ) erPCr) + 0 (rHr».

o

From this it follows, in view of the arbitrariness of E

> 0, that

,.

lim {r-p(r) r-+oo

r

InIF(t)1

~

t

o

dt}~O.

Let us free ourselves now from the restriction that hF(O) = O. For this purpose we construct an auxiliary function W(z) (see § 17, Chapter I) which has no zeros on the positive ray and is such that hw(O) = hF(O). The function ,,," ( ) _ F (z)

Z -- W(z)

'i'

has completely regular growth, and h(O) = O. 'Ve thus find that lim r-P

-

r~oo

(r)

f

In IF (t)1 dt

t

0

=

,.

lim r-P

Cr)

r Inl4>(t)1 t

dt

o'.

+

J r

Inl W(t)1 dt ~ p-1hF(0).

lim r-P (r) r~oo

0

t

By a rotation of the plane one can superpose any ray arg z = () of the angle f3 onto the positive ray, and thus obtain the inequality

oc ~ () ~

r

. r hmr-p(r) r--+oo

In I F (tei~)1

t

dt>p-1hF(fJ)·

0-

It is sufficient to combine this inequality with the inequality (3.06) to complete the proof of the lemma. We note the following consequence of the lemma. COROLLARY. If F(=) is a holomorphic function of proximate order p(r) within the angle oc ~ arg z ~ {3 and of completely regular growth on the ray arg z = e, IX < e < (3, then the follo ..... ing limit exists:

f J}(fJ)'!!.=p-2hF(~)· r

lim r-+oo

r-~(r)

r,

t

(3.05')

A GENERALIZED FORMULA OF JENSEN

Sec. 2]

149

The truth of this assertion follows directly from Lemma 2 and property (5) of the definition of proximate order (Chapter I, page 35). Lemma 2 and its corollary have valid converses, in which one does not have to assume that F(z) is holomorphic.

Suppose that the function F(t) is defined for t

LEMMA 3.

h p (0) =

Furthermore,

if lim

~

0 and that

lim In I F (r) I •

T~OO

rP\T)

r-p("'J~ = p-lhp (0)

(3.16)

,.~oo

or

r

lim

"~.JO

r Jr dtt =

r-P (T)

• o

P

o-'!h (0)

(3.16')

IF'

then the limit . "In IF(r)1 ---''-;-'--'-

I 1m' T~OO

rP (r)

exists. PROOF. The condition (3.16') is a consequence of (3.16). Hence it is sufficient to give the proof under the assumption that condition (3.16') holds. Without loss of generality we shall assume in the sequel that hF(O) = O. Let y be some positive number and E be the set of positive numbers on which

<

InIF(t)1 -TtP(t). (3.17) Let us denote by XE(t) the characteristic function of the set E, and try to estimate the value of the integral r u

JIf

o

ZE (t) t P(t)

~} d: .

0

We have

f If ·IF: ,.

o

l'

.r{f ,.

(t) t P(t)

t!!.} dU"2 t u?"

Q

"

IE

(t) t P(t)

t!!.} du t u

0

T

T

"2

>- In 2

f

ZE(t)tP(tl¥.

u

from which it follows that r

rI r·/

'0

o

,.

u

..F, 0

f 2

(t)

t

It)

'!t!} du 2

U?"

In 2

,.

"4

Z (t) tP\tl ·E

t!!.t

,,.

150

[CHAP. III

FUNCTIONS OF COMPLETELY REGULAR GROWTH

Finally, denoting by m(P) the measure of the set E on the interval (0, r) we obtain

From this inequality and from inequality (3.17), which is valid on the set E, we can conclude that r

f

"

· (t)lnll"'(I)1 {f'l. • .E· t

,.

dt}dU

/_

u ~ o 0 From this inequality and the asymptotic relation In i F (t) I

.,.

2In2m(E 2 )-m(E 4 )rP(1'). i 4P+ 1

< ..t P

r

It)

it is easy to obtain the inequality

f {f l'

o

"

In

1

~ (t) 1 dt} d;

!..

!..

< ~ rP(T)_ i~!:; m (E2) ~ m (E4)

rP(1')+ o(rP(r).

0

Dividing each side by r P('), and taking the limit as r -+ condition of the lemma l'

00,

we obtain from the

l'

lim m(E2)-m(E4) =0. r

1'-+00

From this equation we obtain T

Urn m (E") = 11m m (ET)_m(E2) 1'-+00

r

1'-+00

r

+ lim 1'-+00

r

m (E 2 )=2.. lim m (E\ r

21'-+00

r

and hence we see that the relative measure of the set E is zero. Thus we obtain the result that for every r > 0

IlnIF(r)ll
P (1')

everywhere except possibly on a set of zero relative measure. It follows from Lemma 1 that the following limit exists:

Jim- InIF(r)1 = 1'-+00

o.

rp(r)

This completes the proof of the lemma. Combining this lemma with the corollary of Lemma 2, we obtain the following theorem. THEOREM 2. In order that a function F(z) which is holomorphic al'Jd of proximate order per) within the angle ex <; 0 <; {J (0 = arg z) be of completely regular growth, it is necessary and sufficient that one of the following conditions hold:

Sec. 2]

151

A GENERALIZED FORMULA OF JENSEN

p-thp('J)= lim

p-2hp(IJ) =

r-p(r)}F(IJ),

r ... oo

lim

r-p(r)

,. ... 00

f }~(IJ)dt o

.3b

t

In conclusion we shall establish one more property of the function .rF(O) which will be used in Chapter V. LEMMA 4. For an fntire function f(z) (If proximate order p(r) the function .l}(0) satisfies the inequality IJ;
First of all we note that from the asymptotic inequality In I!(re i &) I

< (hf(/j) + a) rP

(r)

we have the following asymptotic inequality:

},(') < p-t

max [h,(/j)+a) rP(r).

0<8 < 2>t

(3.18)

In the proof of the estimate from below of the function .l}(0) one must distinguish the cases of integral and nonintegral order p. Here we consider only the more complicated case of integral order. The functionf(z) can be expressed in the form

+ a, (..r) zp) In Ia (4: ; p- 1) I+ ~

In I!(z) 1= Re (Pp- t (z)

+

~ I a" I<.r

I

,.I,,: >
In

Ia (a:; p)!.

(3.19)

By Lemma 8 of § 17, Chapter I, the last sum J(z) satisfies, for a sufficiently large the inequality

'T,

lIn I,1 (z)

II < ar?

(r: •

(3.20)

Jb A necessary and sufficient condition of a different kind was suggested by Gol'dberg (it is presented in Azarin (11). He showed that completely regular growth is equivalent to the existence of the limit

for all 01 and p (0 <; 01 < P <; 2'11). Kondravuk (3) proved a similar proposition for meromorphic functions. It is clear that either condition can be taken as the definition instead of the usual one. The advantage of these definitions is the absence of the exceptional sets. Azarin (11) and Agranovic [1) have introduced still another equivalent definition of functions of completely regular growth, one that is useful in the theory of functions of several variables. We state it for functions of a singled variable: a function f(z), holomorphic inside an angle and of proximate order p(r), is called ajunction of completely reguIor growth if the limit lim

"'_00

{J. In1f(rZ)IIp(Z)da(Z)} K

,P(')

exists, where Ip{z) is an infinitely differentiable function with support in a compact set K, and da(z) is the element of area; in other words we require the existence of the weak limit of r-P('lInlf(rz)l. Evidently this indicator, which is called the circuhu indiCalor, is equal to IzIPh(").

152

[CHAP. III

FUNCTIONS OF COMPLETELY REGULAR GROWTH

From the inequality n(t)

< ctp(l) one can quite easily obtain the inequality

f

~,.

, ).

~

_ r~ -1-.. -

Ia I I an I < ~T"

< c ,P I")

dn (t) I.

<

t

(}

(3.21)

(c. is a constant, ;. = 1, 2, ... , p - 1). Furthermore,

(c: is a constant).

(3.22)

From (3.19), (3.20), (3.21) and (3.22) we find that

-- c;r P(r) +

In I f(re i8 ) I>

~

~

I an ' " ~r

We note that

f In I1 r

o

-

t

Idt

- =

lanl t

,. I af" I

In

11 - _z_1 lanl

du In 11 - u 1u

(C; is a constant).

2

>-

f

In 11

-

uI

du u

= -

(3.23)

7

24

'ltl .

o Hence, integrating both sides of the inequality 0.23) we get the asymptotic relation 0

> _(;+'lto~p)rP(").

(3.24)

The relations '(3.18) and (3.24) imply the truth of the lemma. 3. The Basic Theorem on Functions of Completely Regular Order Starting with Jensen's generalized formula and using the established properties (see § 2) of the function JF(O), we can now prove the fundamental theorem on the distribution of the zeros of a function of completely regular growth. We formulate it as follows. THEOREM 3. If a holomorphic function F(z) of order p(r) has completely regular growth within an angle (81, ( 2 ), then for all values of {} and 8 (01 < {} < < (2) except pOSSibly for a denumerable set the following limit will exist:

o

~ s1o.(:1. 0) = lim n (r, 3, 6).

2r.p

.

r-+ oc>

rP

IT)

(3.25)

where 8

sp(~, 6)=(h~('J)-h~(&)+p\1

f hF(ql)d~l

(3.26)

&

The exceptional denumerable set can only consist of points for which h;,(O + 0) rF h;'
Sec. 3J

153

FUNCTIONS OF COMPLETELY REGULAR ORDER

PROOF. First of all we transform formula (3.04) so as to eliminate the derivatives with respect to cpo For this purpose we average each side of the equation with respect to {} and 0, obtaining

8+k &+,

JJ

I~

8

N (r,

r,

I)

dft dij

Il

f

210

rJ~(l)+/).-J~(3~dt}

t

r

=i..{

r

J~(6+k)-JF(fI)dt_ k

o

t

I

f" II

8+k &+1

+ 2;kl.re

8

JJ

In IF (reiw)i

/I

d~ dft dO.

(3.27)

/I

Next we choose the numbers {} and 0 in the formula (3.27) so that the derivatives h~(O) and h~({}) will exist and select numbers I, and k. in such a way that if III < I. and Ikl < k" then

I

hF(O

and

h~(fj)

+ kl- hF(O)

I- - - i

hp(3+I)-hF(&)

1< i

I • h, (tr) < 6'

I

(3.28)

'"

-

Furthermore, if k, and I, are sufficiently small, then 8+k &+1

8

8

lit J f f hF(~)d!pdftdij- f 8

&

hl'('P)d,/<

i·

(3.29)

&

&

Since the function F(z) is of completely regular growth, by Lemma 2 we have for, > '. the inequality

Ir

f r

-plr)

t

J~(O+k)-Jp(O)t!!._..!.. k t pi

h p (0+k)-hF(6)1 k

<~

6•

o

(3.30) It is obvious that the following inequality is valid:

I, r

f r

-p(r)

o

t t h J F (&+I)-JF (&) ~_! ,(&+/)-h,(&) 1 t p2 1

I

t<

..!. 6•

(3.31) Furthermore, if, does not belong to the exceptional set E for the function F(z), then for a sufficiently large '. and for all , > '. we have

f

8

II

&

f

8

In I F (rei,!,) I d -

,.,

(r)

ql

&

h (aI) d

I.!P

I< ..!.

6•

(3.32)

[CHAP. III

FUNCTIONS OF COMPLETELY REGULAR GROWTH

154

From (3.27), (3.28), (3.29), (3.30), (3.31) and (3.32) it follows that for , E E. , > " the following inequality must hold: O+k 3+1

ff

-!...·· IIkr

P \f'l •

N(r,

a,

lj)dadfJ--1-SF(f), 21tp2

3

.J

where III < /, and Ikl < k •. lt is obvious that for / < 0 and k

6)/<1,

>0

f

8+k 3+1

N(r,&,O)~_l_f r P (f')

""

Ikr P (f')

o

N(r, n,lj)d3d6

&

1

<2'1tplISP (f),6)+I.

Taking /

> 0 and k < 0, we obtain in an analogous way the inequality NCr,

a, 6) > _1_ S (n 2'1tp3

r P(f')

F

,

IJ)-I.

From these two inequalities it follows that

r

N (r, &, 6)

1m

,'-+ co

rP (f',

1

n

(3.33)

= .<;n;p "- II SF(", 6).

"EE In view of the fact that the relative measure of the set E is equal to zero, every interval (R, R(I + 15» (15 > 0, R > R4) will contain points which do not belong to the set E. Taking into account the monotone nature of the function N(" D, 0) we find from the last equation that for every' > ' •. 4 and'l < , < 'z, '1' '2 E E

( !:!.)P L('I) (2'1Tp2)-I SF(,c}, 9) _ '2 L(r2) N('2,,c},9) ,p(r)

.;;

'1

e) < N('I',c}, 9) rp(r)

( '2)P L('2) (

.;; N(",c}, 9) ,p(r)

2 -I

)

< -;:;- L('I) (2'1Tp) SF(,c}, 9) + e .

'z

'1

The numbers and can be chosen in the interval,(l - 15) < < , < '. < (l + 15),. Since L(,) is a slowly increasing function we thus have the equation . L (r3) _ 1 )1m ---.

r-+co L (rl)

Furthermore,

Having selected 15 of,

G+ :) < (~r < G+:r.

> 0 sufficiently small, we see that for sufficiently large values

~Pll S, Ca,

0)- 2a

< N~I!; II) < ~pll SFC",

0)+ 21

Sec.3J

FUNCTIONS OF COMPLETELY REGULAR ORDER

155

and hence (3.34) In order to pass from the function N(r, {}, 0) to the function nCr, {}, 0) we note that, for k > I, the fact that the function nCr, .{}, 0) is monotone implies the inequality

f

k··

n (r,3, 6) In k ~ __1_ r P ,rl

-..::::. rP ITI

n (t, 3, 8) dt

t

r

=

N (kr, 3, 6) r P lrl

N (r, -

a, 6)

rP (r)

,

and in view of (3.34) we obtain - . n (r, 3, 6) ./ kP -

11m r-+co

rP(r)

Taking the limit a.;:" -

1

1

~----SF

In k

27t p3

(n"', f'). J

I, we have

f;-

1m r-+co

n (r, 3, ~, ./ _1_ r

Plr)

:::".

2

'lTP

Sf'

(1\ I) 11 ,

J •

n,

6)- N(kr, D, 0),

Analogously, making use of the inequality r

nCr, n, O)lo!

~

f

n(t'tf), 6) dt=N(r,

kT

which holds for 0

< k < I, we find that . n (r, 3, 8)........ _1_ (1\ 6) \ 1m ) -? SF v, • r Plr 2'ITp

r -+ co

Thus the limit (3.25) exists for those values of {} and 0 for which the derivatives h'({}) and h'(O) exist, i.e., for all except possibly a denumerable set. This proves the theorem. COROLLARY. In order that the set oJzeros oj aJunction oj completely regular growth may have a density within some angle (at, fJ), it is necessary and sufficient that its indicator be sinuSoidal within this a,.gle.

Indeed, if the indicator is sinusoidal, then Seat, fJ) == 0, and in view of (3.25) the density of the set of zeros is equal to zero. Conversely, a zero density of the set of zeros implies that s( at, fJ) == 0, and by property (g) of the indicator (Chapter I, page 56), it will be sinusoidal. From the theorem just established, and from Theorem I, Chapter II, it follows that for nonintegral p, the class of entire functions with regularly distributed zeros and the class of functions of regular growth coincide.

156

FUNCTIONS OF COMPLETELY REGULAR GROWTH

[CHAP. III

We shall prove that this is also the case for entire functions /(z) of integral order. By Theorem 3 the set of zeros of the function/(z) has a density. But in this case, in line with the definition of a regularly distributed set, one must prove also that the foUowin~ limit exists:

}~"!., L ~f) {a +

+~

a;p},

lanl<:r

where a is the coefficient of the term of degree p in the polynomial P(z) of the expression (1.38) for the function /(z); the an are the zeros of this function, and L(r) = rP(r)-,. In order to prove the existence of this limit we form the function q'r(Z)=

II

0 (: ;

Ian I <: r

r-- I )

n

.

II

I Un I > r

0 (:;

p).

n

By Lemma 5 of Chapter II, for any r not belonging to some EO-set EtII , the following limit exists: lim In I
(reiGU == hel> (fJ),

rP lr)

Furthermore, since /(z) is of completely regular growth, we have the following result, if r E E"

Noting that

Ll
a;P)e ifG }

lanl
_

In If(re iG ) I _ In I II>

-

(fe')'& I +

r Plr)

rP1r!

we find that for arbitrary () and r not belonging to the set EtII

(I) 0

I

+ E, the

limit

rl~~ L:f)Re {(a++ ~ a;p)eiP&} =h,(6)-h.(6) lanl
exists. Taking first ()

= 0, and then () = 1T/2p, we establish the existence of the limit 1 {a + P 1 ~ ~ a:p} (3.35) (J,~ = I'1m L (f) .. r-+oo

lalll
ifreE, + E tII , Let us free ourselves now of the requirement that r e E, purpose we estimate the quantity 1 L(f)

~

~

r
I alll- P

+ E tII .

For this

Sec. 3]

157

FUNCTIONS OF COMPLETELY REGULAR ORDER

Expressing it as a Stieltjes integral, we obtain

~

1

L (r)

lanl- P

4J

r < Ian I<:kr kr

__ 1 -

L(r). r

rdn7(t) -_

ICr

1 [n (kr) L(r)

(kr)P

n (r>]

-7

+

p

L(r), r

r n (t) dt. tP+1

The first summand on the right hand side may be written in the form

[n (kr)

1

(kr)~

L (r)

n (r)]

n (kr)

- 7 = (kr)p \krl

L (kr) L (r) -

n (r) rP (rl'

It will tend to zero as r - 00 if the density A of the set {a,,} of zeros exists. The last summand satisfies the asymptotic inequality

kr

_P-fn(t)dt< L (r) t P+ 1

(A + £) 2"

L (r)

r

<

k"

St P(tl- P- l dt

p

( A +~)p 2 (k

L (r)

r kr

r

t(kr)frp-ldt=(!l+~)L(k") (kP-l) .

2

L (r)

•

r

Finally, we obtain the asymptotic bound /(r)

~

lanl-P«l+a)(kP-l).

r< lanl<:kr

(3.36)

For an arbitrary k > 1, and for sufficiently large r, every interval (r, kr) contains a point rl that does not belong to the set EI + E.. In view of (3.35) we have the asymptotic inequality

or

I

L;r)(a++

~ o;p)-o, 1<;,

Ian I<:r,

since lim r - 00 L(r)/L(r1 } = 1. Furthermore, from (3.36) we see that for k sufficiently near unity, the following asymptotic inequality holds:

I

PL\r)

~

r
o;P I< ;.

FUNCTIONS OF COMPLETELY REGULAR GROWTH

[CHAP. III

Therefore

It follows that the limit " I'1m -L( 1 ( a+1 0,= r-+oo r) P

(8)

l: lalcl ",r

-1) aP k

exists if r goes to infinity in an arbitrary way. Combining this result with Theorem 2 of Chapter II, we obtain the next theorem, which may be considered to be the fundamental theorem of the theory of functions of completely regular growth. THEOREM 4. In order that an entire function F(x) ofproximate order per) be a function of completely regular growth, it is necessary and sufficient that its zeros be regularly distributed.

In Chapter II there was given (in Theorem 4) an asymptotic representation of entire functions with regularly distributed zeros. From this representation, and from the theorem just proved for functions of completely regular growth, it follows that for all but a denumerable set of values of 0 the limit kF(fJ) =

'0

lim arg/(re' ) r-+oo

r Plr )

exists. The same result can be obtained for functions of proximate order per) and of completely regular growth within some angle; however, we shall not dwell upon this matter.' REMARK. From condition (8) it follows easily that the canonical representation (1.38) of an entire function f(;:) of completely regular gro ....th of integral order p and of the normal type can be ....ritten in the form

where m is a non-negative integer and Pp(z) is a polynomial of degree not greater than p. In particular, an entire function of exponential type and of completely regular growth admits the representation

f • Sec A. Pftuger [3].

(z)

= Zllle aHb

lim

n(

R -+ 00 I ale I < R '

1_

-=-). ale

Sec. 4]

159

INDICA TOR OF THE PRODUCT OF TWO FUNCTIONS

Indeed, an entire function is representable in the form

n

• - f - ...

/(z)=zmeP(Z) lim (1_-=-)I/ k R ... 00 I a k 1< R ak

zp-l

+

zP

+_

(p-l)a?k- 1

pa~

'

that is, as

/(z) = zme P(.) i~"!x, exp

I(

~

ak"P) zP)

lakl
n

z

zp-l

- + ... + ------:-

X lim (1_-=-)e ak R... 00 I ak ! < R ak

(p-l)a£-J

In order to obtain from this the required representation of the functionf(z) one needs only to use condition (B). 4. Indicator of the Product of Two Functions In § 15 of Chapter I we mentioned that the indicator of Ihe product of two functionsf(z) and rp(z) holomorphic within an angle (01' (2) (ioes not exceed the sum of the indicators of the factors, i.e.,

(3.37) The same relation obviously exists for the generalized indicators if one assumes that all three indicators have been computed for the same proximate order. We shall show that if one of the factors has completely regular growth. then the equality holds in (3.37). Indeed, from V. Bernstein's theorem (Theorem 31, Chapter I) it follows that the inequality

° °

holds for arbitrary fixed (01 < ~ (2) on a set E' of positive upper relative measure. If one of the factors (for example, rp(z» is a function of completely regular growth, then for all values of r > 0, except possibly for a set EO of zero relative measure, we have the inequality In I ~ (reiD) I> [h, (I}) -. e) rPI"). Thus for all values of r belonging to th~ set E' - EO. E' of positive upper relative measure we have In I~ (reiD) / (reiD) I [h f (6)+ h~ (&) - 2eJ rPlr). (3.38)

>

Combining (3.37) and (3.38) we obtain the following theorem. THEOREM 5. If the functions f(z) and rp(z) are holomorphic within the angle (01' (2) and if one of them is of completely regular growth, then the indicator of the product of these functions is equal to the sum of their indicators.4a 4a Azarin [1] has proved the following converse theorem. If .f(z) is hohJmorphic and har proximate order p(r) iMide an angle (II •• llUo and if for every hoIomorphic jrmction .,(z) of proximate order p(r) the indicator of the product j{z).,(z) equals the .rum of the indicaton, the,. j{z) is of completely reguku growth.

FUNCTIONS OF COMPLETELY REGULAR GROWTH

160

[CHAP. W

Let US also note that if both factors are of completely regular growth on some ray, then the product is of completely regular growth on that ray. If the entire function feZ) satisfies the condition •

11m

In M, (r) rP

1'-+00

t,.)

=0.

then it follows from Lemma 4, Chapter I, that .

n,(r)

11m ---;trI = 0,

1'-+00

r

and hence the set of zeros of this function will have zero angular density. If p = lim,_oo per) is an integer, then it follows from Theorem 18 of Chapter I that "

0,=

.

S,(r)

hm L-() =0. 1'-+00 r

Hence, by Theorems I and 2 of Chapter II, th~ functionf(z) has completely regular growth and its indicator is equal to zero. Thus by Theorem 5 we obtain the following result. COROLLARY l. within an angle (° 1 ,

If F(z) ( 2 ),

is a holomorphic function of proximate order per) while 'F(z) is an entire function such that .

M,(r)

hm --(-=0,

1'-+co r P '"

then the indicator of the product of thest' functions within the angle (° 1, 02) is equal to the indicator of the function F(z).

From Theorem 4 it is easy to obtain the following corollary. 2. If the functions fez) and 'F(z) are holomorphic within an angle if 'F(z) is of completely regular growth for some p(r), and if

COROLLARY

(° 1,

( 2 ),

lJ!'(z)

=

fez) ,(z)

is a holomorphic function within the same angle, then

To derive this equation one considers the product IP
Sec. 5]

CASE OF A SINUSOIDAL INDICA TOR

161

5. Some Consequences of the Generalized Formula of Jensen. Case of a Sinusoidal Indicator In § I we showed that if an entire function has completely regular growth on every ray of some dense set of rays, then the function is of completely regular growth in the whole plane. It is not difficult to construct an example of an entire function which has completely regular growth on a finite set of rays (arg z = br/n, k = I, 2, ... , 2n; n arbitrary) but which is not of completely regular growth in any angle greater than 'IT / n. Indeed, let p(r) be an arbitrary proximate order, and let rk be a root of the equation rP(r)=k (k=l, 2, ... ). Let n be an integer, n

> p.

The set of zeros of the entire function 00

cI>(z) =

II(I- ~z:) k=1

has an angular density. Furthermore, /S~ = 0. Thus (f)(z) is of completely regular growth. Let us mark the points r k lying in the intervals (22m - I , 22m) (m = 1,2, ... ). Next we renumber them in the order of increasing moduli, and denote them by {ap}. Analogously, we denote by {h p} the set of points rk lying in the int~rvals (22m, 22m +l), and we set

II(I - a,z:r ,=1 II(I + b,Z;:)3. 00

W(z)

=

00

,=1

The set of zeros of the function 'Y(z) does not have an angular density in any angle containing a ray of the set arg z = br/n (k = 0, I, ... , 2n - I), and therefore 'Y{z) is not a function of completely regular growth in anyone of these angles. However, on each ray of the type arg z = (2k + 1)'IT/2n(k = 0, I, ...• n - I) we have II¥(z) I = I cI> (z) I. and therefore the function 'Y(z) is of completely regular growth on each one of these rays. The function 'Y(z) thus serves as the required example. The situation is quite different if one requires additionally that the indicator of the growth of the function j(z) be sinusoidal within the angle [ex, P], i.e., if

"r(lj) =

AcoslJ +8 sin a

In this case, for the investigation of the behavior of the function within the angle [ex, Pl, we transform formula (3.04) by' integrating both sides of the equation

162

FUNCTIONS OF COMPLETELY REGULAR GROWTH

[CHAP. III

with respect toO from ffl to fP2' and with respect to 0 from 'f2 to '1:\ (IX q2 < q 3 < (j). Thus we obtain the equation

f

If.

'h

2-:rS "

NF(r, 1\, O)dfldlJ

~I

I, l'

=

< q1 <

t t t dt (JF(~:I)(~2 -~t)+ J F(!P2)(!Pt-~3>+JF(!Pt)(!P3-~2)) t

o If,

to,

+ Sf '
'1'.

a

JI

d~ dll d:J,

In F (rei,) I

(3.39)

"

For the purpose of estimating the value of the right hand side of (3.39) we note the asymptotic relation In IF (tei'f) I

This implies that for p J F(~) =

f

l'

< [hF(fP) + ~] t P

(t)

(a> 0).

> 0 we have the asymptotic inequality

In I F ~tei'l') I dt

< 0 (rP (Ti)+[hF(~)+-;-]

o

f l'

tP(t)-l dt

0

From this it is easy to see that

f J~(~) ~ < l'

lim ,-P (T) 1'-+ uo

t

o

p-2hF (fP).

(3.40)

Furthermore, Lemma 2 implies that if F(z) is of completely regular growth on the ray arg z = fj2' then l'

lim

,-p(T)

1'-+00

~t

JJ;'('1I2)

=p- 2 h,(fP2)'

From (3.39), (3.40) and (3.41) we find that for every given ~3

Jim 27tr- p (T1 f 1'-+00

'h

(3.41 )

0

't

E

>0

fNF(.·, fI, fJ)d:!dfJ '1'.

+

. ffJ ., ..

hF(?)d?dfldIJ.

,.

'F.

•

(3.42)

Sec. 5]

163

CASE OF A SINUSOIDAL INDICA TOR

If in the left hand side of the equation we replace the upper limit by the lower one, then the sign of equality in (3.42) will apply only for functions of completely regular growth. This we shall show later. LEMMA

IX

(a

5.

< fli2 < 13),

If

the function F(z) is holomorphic and of order p(r) when and is of completely regular growth on some ray arg z = ~2 and if for a < fIi) < cP < fli3 < 13

< arg z < fl,

'f>

21t lim r-p(rlf

'h

f

Nj.(r, 0, fl)dfld'J

r~co

=

p-2 {h F (
+f '1'0

8

'1'1

f

fhF(If)d
'PI

II

then F(z) is of completely regular growth within the angle

IX

< q; < fl.

PROOF. First of all we note that in view of (3.42) one must replace the lower limit simply by the limit in the formulation of the lemma. From equation (3.39) and the hypotheses of tke lemma, we obtain r

f

}~moo {[,-p(r) J~(
r

+ [r-p(rl

f ~(lf2) J

d: - P-2 hF(
o

r

+ [r-p(rl .f J~(
+f '!'.

"

•

'!'l

II

J

JlhF.r(
From (3.40), (3.41) and the asymptotic inequality

(e> 0) it easily follows that the upper limits of all four summands are non positive. Transposing all but one of the summands to the right hand side, we see that the lower limit of this summand is non-negative.

FUNCTIONS OF COMPLETELY REGULAR GROWTH

164

[CHAP. JII

Hence each of the summands must have the limit zero, i.e., l'

lim rP (1')

1'-+00

rl~'"a,

••

'ft

f

f

••

••

fJ~(IP) t!!.t = p-2hF(~)'

•

o

•

f[hF.1'(tp)-hF('P»)d'Pd3dO=O I)

By Lemma 3, it follows from the first equation that F(z) is of completely regular growth within the angle IX < arg z < p. This proves the lemma. We now state a theorem on functions with sinusoidal indicator. THEOREM 6. If the JUllctioli F(z) is holomorphic and oJ proximate order per), if it has within the angle IX < arg z < p a sinusoidal indicator, and if the Junction

is oJ completely regular growth on some ray within this angie, then the Junction is oJ completely regular growth within the whole angle. PROOF. We begin with the inequality (3.42) inequality can be written in the form

p-2

where

f':I f" [sr(8) -

"'_

The right hand side of this

sF(9») d& d'J,

'II.

.p

SF(!P)=h~('P)+p2f h(~)d~. For a sinusoidal indicator, S/i'(q?) = const and the right hand side of (3.42) is equal to zero. Since N(r, D, 8) is a non-negative number it is obvious that 'f:a

f f N(r, n, 0) dS db = O. If3

lim r-P (r) 1'-+00

If:!

If.

The conditions of Lemma 5 are thus fulfilled and F(z) is therefore of completely regular growth within the angle (IX, Pl. We note that for a sinusoidal indicator, (3.25) and (3.26) imply that lim n (r, &,~ = O. r-+(.I)

rP (f')

This completes the proof of the theorem. The requirement that the function be of completely regular growth on some ray within an angle in which the indicator is sinusoidal may be relaxed. One has the following theorem. THEOREM 7. If for a < fJ < {J a holomorphic junction of proximate order p(r) has a sinusoidal indicator within the angle a < fJ 1 <; fJ <; fJ2 < {J, and if in addition the function is of completely regular growth on the ray arg z = tJ 1, where hF
Sec.5J

165

CASE OF A SINUSOIDAL INDICATOR

From formula (3.39) it follows directly that

PROOF. r

S(J F(~a)('f2 -

t

t

t

dt
o e.

..,

8

+S

S

flnIF(rei'F)ld'fd!ldO~O

,.

,.

8

We rewrite this inequality in the form r

(
f

f r

dt

t

Jp(?s)T~(
o

t dt JF(!{I2)T

0

r

-(?s-?2)fJ~(
c.

"'.

ff '!'.

'F.

8

Sln/F(rei'l')/d
Let us set flJz = 01 , 01 < flJa < O2 , and at < fIJI < 01 • Since the function F(z) is of completely regular growth on the ray arg z = CP2' we may apply .the corollary to Lemma 2, obtaining r

(?2-?1) lim

r-p(")

r-+ao

fJ~(
0

-

lim r-+ao

Together with the asymptotic inequality r

r-p(r)

S

t

dt

Jp(?1)7

1 < pa-hp(
o this implies that (!{I2 -

r

!{Il) lim r -+ ""

r-p(r)

SJ ~ (
~~ Ihp(

.

., ,. 8

- ,.f ,.f

.fhp(
[CHAP. III

FUNCTIONS OF COMPLETELY REGULAR GROWTH

166

This last inequaltiy is valid for arbitrary 9/1 it by ipl - ({II' and taking the limit as q 1 -

,.

< 'f 2' and r2 < ra < O2, r2'

Dividing

we obtain

f

t dt ~rp,,) • Jp('~:1)T ,. .. ""

0

", . 8

> ;2 {hp('~2)+h~(92-0)(9:1-Ifll)-P~

JJ hF(~)d9d·J}. ~.

From the condition h~(ip2

+ 0) = h~(T2 -

Since by hypothesis h p (9/) yields

= A cos P(T -

.

11m

0), we see that

To) when r2

~

T

~ ()2'

this inequality

,.

-r

r-?(').

,·4000

'h

t dt 1 J F (9a}-t-> p:l h p (9;l)'

0

Combining this with the inequality (3.06) we easily see that for 01 the following limit exists:

~

T

~

O2

It follows now from Theorem 2 that F(z) is of completely regular growth within the angle 01 ~ arg z ~ ()2' The theorem is thus proved. Formula (3.39) permits us to draw some other conclusions about the distribution of zeros on rays of completely regular growth. Let us denote by p(t, ip) the number of zeros of the function F(z) on the ray arg z = T whose moduli (absolute values) do not exceed a number t > 0, and let us consider the integral ,. P(r. If) =

It is obvious that for {}

< T < ()

J

P (t. If) dt

--t-

.

o

N(r, U. 'J)'~ P(r, cp).

If the function F(z) is of completely regular growth on the ray arg z

=

ip, then

Sec. 5]

167

CASE OF A SINUSOIDAL INDICATOR

setting 9'1 = 9' - k, 9'2 the inequality k'llim r-+

p(r'(r~) rP

OCI

!

= 9'. 9'3 = 9' + k (k > 0) in formula (3.42) we obtain

-
'9+ k

co

8

+f, f f h ('l:)d'l:dl}d6. ",-k&

This yields directly the next theorem. THEOREM 8. If F(z) is a holomorphic function of proximate order p(r) within an angle containing the ray arg z = 9', and if the function F(z) is ofregu;ar growth on this ray, then

r

r~~

P(r, cp) rP(r)

1

,

-< 27t02[hF (~+O)-h~(
We might recall that for functions of the first order the quantity h~(9' + 0) 0) has a simple geometric meaning. This quantity is equal to the length of a straight line segment of the boundary of the indicatl Ir diagram whose outer normal has the direction of the ray arg z = 9'. We note also that the theorem remains true if we understand by per, 11') the number of zeros in the intersection of the circle Izl <; r with some region which lies almost entirely5 within an arbitrary angle containing in its interior the ray arg z = 9'. For example, such a region may be a parabola with axis directed along this ray. The requirement that the function F~z) be of completely regular growtt:. ~n the ray arg z = 9' is essential. In Chapter V we shall construct a function of exponential type and of nonregular growth for which h~9' -

,- ;- P (r, J) 1m

r-+ OCI

r

=

I

and for which the boundary of the indicator diagram does not contain a segment of a straight line parallel to the imaginary axis (Chapter V, §S). , The term "almost entirely" is to be finite region.

unde~too4

here in the sense: entirely except for a

CHAPTER

IV

UNIQUENESS, INTERPOLATION AND COMPLETENFSS In this chapter we apply the apparatus developed in the preceding chapters to problems about the uniqueness of entire functions, the interpolation of entire functions and the completeness of systems of functions in the complex domain. These questions are, as we shall see, closely connected with each other. There is also a connection between interpolation and the problem of the dependence of the asymptotic behavior of an entire function in the whole plane on its asymptotic behavior on sequences of points tending to infinity. The first three sections of this chapter are devotee! to questions of uniqueness. The general problem of uniqueness can be formulated in the following way. Let the holomorphic functionf(z) belong to a linear class (for example, let it be at most of order p and normal type). For which systems of linear functionals Aft (n = 1,2, ... ) will the values a .. = An(f) determinef(z) uniquely?l As functionals we may take, in particular, the values of the function at certain points, i.e., A ..(I> = f(z .. ). A set {z .. } such that when two functions of a class coincide on the set they are identically equal will be called a set of uniqueness for the given class. By a familiar theorem of the theory of functions, a set which has a limit point in a region G is a set of uniqueness for all functions that are holomorphic in the region. Hence for entire functions the interesting sets of uniqueness are those with a limit point at infinity. The corresponding uniqueness theorems state that a function of "not too rapid growth" cannot vanish on a "sufficiently dense" set unless it is identically zero. Thus these theorems can be considered as generalizations of a familiar theorem on polynomials: a polynomial of degree at most n which is not identically zero can not have more than n zeros. Theorems of the same character also hold for functions holomorphic in an angle. A typical uniqueness theorem of this kind is the following theorem of Carlson [I], which plays a prominent role in interpolatory questions.

If a holomorphic function fez) of exponential type in the half-plane Re z > 0 vanishes at the integral points n = I, 2, 3, ... and the width ht<:",/2) + h,( -."./2) of its indicator diagram is less than 2.". then fez) == o. 1

This formulation of the uniqueness problem was given by A.

GonQrov [II.

168

o. Gel'fond [2J and V. L.

INTRODUC110N

169

This theorem is sharp, since sin 1TZ vanishes at all the integers and the width of its indicator diagram is exactly 21T.2 In §§ 2 and 3 we shall prove considerably more general theorems of the same type. I n most of these theorems the class of functions of completely regular growth plays a fundamental role. They represent the class of "boundary functions," which have the greatest density of zeros for a given rate of growth. To clarify this fundamental role we mention Theorem 3 of § 2.

The set of zeros ofevery entire function, not identically zero, ofproximate order p(r) satisfies the inequality

There is equality for entire functions of completely regular growth, and only for such functions. This characteristic extremal property of entire functions of completely regular growth could serve as their definition. Therefore entire functions of completely regular growth are the functions with given proximate order and given indicator which have the greatest lower density for their sets of zeros. We shall prove analogous theorems for functions of finite order which are holomorphic in an angle. In § 3 we present theorems in which the set of uniqueness is either a linear set or a set of points with an angular density in an angle. Along with the question of the unique determination of an entire function by its values on a set of points, it is natural to consider the question of reconstructing the function from these values. This problem is solved by interpolation series. Here we shall study the question of the representation of an entire function by a Lagrange interpolation series, where we choose the nodes to be an R-set (Chapter II, § I). Under this restriction we shall obtain definitive results. The problem of the uniqueness of entire functions is connected with the problem of completeness of systems of holomorphic functions in a region. This • To prove this theorem it is enough to remark that ifI(z) satisfies the hypotheses and is not identic:ally zero then the quotient (z) = z/(z) , slnu is a function of exponential type in the hatf-plane Re z

> 0 with indicator

h. (8) = h,(8) - -\ sin 81· Therefore

h.

(i)+h. (-i) <0,

which contradicts property (h) of the indicator (d., § 16, Chapter I).

170

UNIQUENESS, INTERPOLATION AND COMPLETENESS

[CHAP. IV

connection is investigated very fully in a paper of A. I. Markusevic [2]. He established that every theorem on uniqueness implies a theorem on completeness, and conversely. In § 6 we present theorems on the connection between completeness and uniqueness, and in § 7, by using this connection and the uniqueness theorems of the first few sections, we obtain some theorems on the completeness of systems of functions. 1. Uniqueness lbeorems for Entire FunctioDS of Finite Order Uniqueness theorems can be formulated as indicated above, i.e., under restrictions on the growth of a function that is entire (or holomorphic in an angle) the vanishing of the function on a set of large density (ordinary, upper or lower) implies that it vanishes identically. Evidently we can give an equivalent formulation of each such theorem, namely that for a function that is not identically zero the density of the set of zeros can not exceed a number that depends on its growth. It is natural to obtain such estimates for the density of zeros of an entire function of finite order from Jensen's theorem (cf. Lemma 4 of Chapter I). THEOREM !. Let f{L} be an entire function of proximate order p(r) (p > 0), not ide.'llically zero, n,(r) the number of zeros (not at the origin) off(z) in a disk of radius rand

r

' n (/)

N,(r)

= .J + d t .

(4.00)

o

Then

J' h,(f)d'J. 2"

1 lim -(r-<:r-+ 00 r P ) 21t -

PROOF.

N,(r)

(4.01)

o

By Jensen's theorem 3 we have the equation

f h

Nt (r)

1

- = 21t rP(r)

o

h,•r (O)dl) -

In 1/(0) I r PIr)

•

(4.02)

The right side of this equation is easily estimated if we use the asymptotic inequality (4.03) which was proved ir. Theorem 28 of Chapter I. From (4.02) and (4.03) the conclusion of the theorem follows at once. • If/(0) = 0, we apply Jensen's theorem to /(z)/z-, where m is the multiplicity or the zero at O.

Sec. 1]

171

ENTIRE FUNCTIONS OF FINITE ORDER

THEOREM

2.

Under the hypotheses of Theorem 1 we have the inequality

(4.04)

From the monotonicity of n,(t) we easily get the inequality

PROOF.

(4.05)

from which we obtain (4.06)

Inequality (4.04) follows directly from (4.06) and Theorem 1. We now show that (4.04) is sharp; indeed, we show that there is an entire function for which there is equality in (4.04). In constructing such a function it is helpful to remark that there can be equality in (4.04) only for functions of very irregular growth. In fact, there is equality on the left of(4.05) only whenf(z) has no zeros in the annulus r < Izl < el/Pr. There is equality on the right of (4.05) only when f(z) has no zeros in the disk Izl < r. Hence equality in (4.04) is possible only when n(r) increases only on short intervals which are sparsely distributed along the positive real axis. After these preliminary remarks we turn to the construction of a function for which there is equality in (4.04). We consider only the case of entire functions of normal type of the ordinary order p. EXAMPLE.

We consider the sequence of integers np

=

22 • and construct

the function (4.07)

For nl'p

< r < nlil we have n(r) = 2'3 +2 4

where EJ: < (k· Hence

+ ... + 2~ =

nk(1 +ak).

1)2-z-t-1 • (4.08)

172

UNIQUENESS, INTERPOLATION AND COMPLETENESS

On the other hand In If (z) I =

[CHAP. IV

nk

~ In 11 - ( z.!. )~111 + In 11 - (:: fP- i nP

11= 1

11

+ ~

In

11 _( Z.!..)nl1j. nP

lI=k+l

For nl'p

(4.09)

11

< r < nl'-!\ we obtain

P

< 2nk_lln 2r < 2r21n '2r = If P ~ k

o (r P).

+ 2, we have asymptotically rn-I/p < ! and from the inequality Iln(l-u)I<1IU11ul <2Iul.

which holds for

lui < t, it follows that

Combining (4.09), (4.10) and (4.11), we see that asymptotically for

r

(4.10)

< nl'-l'l In 1/ (z) I = In

or for nt(l

n" /1 - ( ~:) p I+ 0 (rP).

+ e) < r < nt+l (e > 0) In I/(z) I =

nl'p < (4.12)

P

,p

n: In ~n" + 0(1).

pro

(4.13)

The maximum of the first term on the right side of this equation is attained for r P = ent and consequently h, (IJ) --

"

-I' In I/(re') 1_ ,

1m r -+00

rP

--.

pe

(4.14)

Combining this with (4.08), we obtain 21<

lim n (r) = r-+ co

,p

~

rh,(fJ) d.~.

271: • o

i.e., (4.04) becomes an equality for the function we have constructed.

(4.15)

Sec. \]

173

ENTIRE FUNCTIONS OF FINITE ORDER

In estimating the lower density of the set of zeros of an entire function a fundamental role .is played by functions of completely regular growth. We have the following theorem. THEOREM 3. The lower density of the set of zeros of every entire function f(z), not identically zero, of proximate order p(r) (p > 0) satisfies the inequality

f

2K

.

_11m

r~ao

nr(r)

r

JI \r)

p

-< 21t u

h, (0) diJ.

(4.16)

There is equality for functions of completely regular growth and only for such functions. c PROOF. The first part of the theorem follows immediately from (4.01) and the inequality

nr(r)

11m f"(f=) r-+ao r

. Nr(r) -< pr:+oo Itm r,·r

(4.17)

- - I- ) •

The latter inequality follows from property (4) (cf. Chapter I, § 12) of proximate orders. The essential part is the second statement of the theorem. In Chapter III (Theorem 2) it was proved that the set of zeros of a holomorphic functionf(z) of finite order and completely regular growth in an angle has an angular density, and that

r1m

r~ao

f•

a,

nr(r, 0) _ _ 1 [h'«(j)-h'(&)+ 2 P (r) 27tp' f P r . •

h ( )d ]

,IF

IF.

Taking {} = 0, (J = 211', we have

It remains to show that there is equality in (4.16) only for entire functions of completely regular growth. It follows from (4.17) that when there is equality

• The estimates (4.04) and (4.16) were essentially given by R. P. Boas. cr., the book [I). In a recent paper R. C. Buck (1) gave the following estimate for the distribution of the zeros of a function of order p and normal type «1:

• 11m

II

(r)

r-+ao"

+

Jim n (r) "" ea,:lJ. r-+ u.l

rP

A. Pftupr (I] gave a theorem c:IoIely related to Theorem 3.

174

UNIQUENESS, INTERPOLA nON AND COMPLETENESS

[CHAP. IV

and since by Theorem 1

we obtain

f

2"

N,(r)

lim --(,.P r

,. -+ 00

)

=

1

27t

h, (0) d'J.

o

Combining this equatiotl with Jensen's theorem (4.02) we obtain 2"

f (h,

lim

h,.,. (6») dO

(IJ) -

,.-+00 0

and by using the asymptotic inequality h, .•(O)

=

0

< h,(O) + e (e > 0) we infer that

2"

lim

f/

hr.,. (IJ) I dfJ = O.

h, (IJ) -

,.-+ co 0

(4.18)

To deduce from this thatf(z) is of completely regular growth we first use Theorem 6 of Chapter II, according to which the family of functions h, .•(O) is equicontinuous for all positive r except possibly for a set E of values of arbitrarily small upper relative measure. We shall show, by using this theorem and (4.18), that for r E E and r -+ co the function h, .•(O) tends uniformly to h,(O). In fact, for sufficiently small 15, Ikl < 15 and r E E 9+k

/hr.r(IJ)-h,(fJ)/<

£

1

r /hr.,.(
2+/i , 8

21<

f

< -; +ri1 !h,.r(
> r.

o

(4.19) Hence we have shown that for arbitrary e > 0 and 1] ~ 0 the inequality (4.19) is satisfied for all positive r except perhaps for a set of upper relative measure less than 1]. By Lemma 1 of Chapter III it follows from this that f(z) is of completely regular growth. The·theorem is established. REMARK. For p = I the integral on the right of (4.1 6) is equal to the length of the boundary of the indicator diagram of f(z).

2. Uniqueness Theorems for Functions of Finite Order, Holomorpbic iD an Angle In this section we obtain uniqueness theorems for functions that are holomorphic and of finite order in an angle. These are analogs of the theorems for

Sec. 2]

FUNCTIONS OF FINITE ORDER, HOLOMORPHIC IN AN ANGLE

175

entire functions that we established in the preceding section. In Chapter III, by using the generalized Jensen theorem, we obtained estimates for the density of the set of zeros of a function F(z) in an angle (inequality (3.42) and Lemma 5) analogous to Theorems I and 3 of the preceding section. However. these estimates were obtained under the very restrictive supplementary hypothesis that F(z) has completely regular growth on a ray inside the angle. In this section we shall prove further uniqueness theorems for holomorphic functions of finite order in an angle without making any assumptions about the regularity of their growth. For this purpose we shall develop a new formula analogous to Jensen's. We note that without loss of generality we may suppose that the angle in which F(z) is given is the half-plane Re z ;> 0, since the general case can be reduced to this by the transformation w = ei%zY (y > 0,0 " at < 211"). We now establish the analog of Jensen's formula for a half-plane. Let F(z) be holomorphic in the half-plane Re z ;> 0 and let F(O) #- O. Take a number d such that F(z) has no zeros in the closed disk Izl "d. We construct a contour r d, R consisting of arcs of the circumferences Izl = d and Iz - RI21 = RI2 (R > d). With each zero of F(z) as center, draw the circumference Cj of radius 15, where 15 is chosen smaller than the distance from rd,R to the set of zeros that lie inside this contour, and smaller than the distance between the zeros. In the region bounded by the contour r d,n and the circumferences Cj , consider the two harmonic functions u(z) = In IF(z)1 and v(z) = l/R - cos Olr (z = rill) and apply Green's theorem to them. We obtain

f (U::-V::)dS= ~ J(U::-V:~)dS. J (ei)

(T", B)

Taking the· limit as 15 -+ 0 and noting that v = 0 on the circumference Iz - R/21 = R/2, which we denote by r R' we obtain for large R

~-.(B) 2

f

1

=2K

In I F (R cos De ia ) I:: R dD

+ 0 (I),

-~+.(B) 2

where Zj = are the zeros of F(z), at(R) = arc sin dlR, and 0(1) is an integral of a bounded function along an arc of Izl = d. Noting that rj cos-1 OJ is equal to the diameter R j of the circle r R about the zero r j ei8j , we can write the left side of the equation in the form j r j ei8j

B

'\""

~ rJ
(1 R

cos 8i )

---rj =. 0

r(/i-T 1

1)

dv,(t).

176

UNIQUENESS, INTERPOLATION AND COMPLETENESS

[CHAP. IV

where ."".(1) is the number of zeros of F(z) inside r t • Transforming the right side of the last equation by integration by parts, we obtain

~

(~ -

B

cO:JOJ )

"J
dt.

=

f

II R2 cos 2 8 on

r. R' we obtainli

a--(Bl

B

ftr

f vp~t) 0

On the other hand, noting that dvldn vp(t)

=

dt

o

=

I 21t

In / F(R cosOe") / dO R cosIO +0(1).

(4.20)

K

-"i+_(B)

Equation (4.20) is an analog of Jensen's formula and provIdes a means of establishing uniqueness theorems for functions holomorphic in a half-plane. We shall give another derivation of this formula based on the same principle as the derivation of Jensen's formula in Chapter I, or of the generalized Jensen formula in Chapter III. Let r t denote the circumference Iz - 1/21 = 1/2, and choose t so that this circumference does not pass through any zeros of F(z). We have vp(t)

=~

f

d [arg F(z)].

(r,)

The transformation

I

111=-

takes

r t into the line II = vp(t) =

z

1/1 of the w =

f""!

~

[arg

II

+ ;v plane, and we have

F(U~IV)L,,'-l dv

-00 00

=~

f:U [lnIF(u~'V)lt=t_ldV. -00

Integratmg both sides of this equatIon with respect to rl and Jrl we obtain

II

between the limits

• If we analyze the term 0(1) when F(O) = I, it is easy to sec that as d tends to l Re F' (0), and taking the limit in (4.20) we obtain the equation

-+ 0

the term 0(1)

" "i

__ I f In/F(RcosO..a)/ dll -~ReF'(O). f. vp(t) fA dt - 2x R cos 8 2 B

••

o

-.

l

..

Here the intcgral on the right side of the equation is to be taken as a principal value (sec Levin [14D·

Sec. 2]

FUNCTIONS OF FINITE ORDER, HOLOMORPHIC IN AN ANGLE

Noting that on

177

r, we have v = - r 1 tg8,

z=tcosUe!",

dv=-~

t cosll6 '

we find that

r R

'Ip (I)

· i 3 dt

•

"2" 1 • -:be • (R-lln I F(R cos Uejfl) 1-1- 1 In IF(£ cos 8e fll ) I) ~8 cost

r

Furthermore, it is easily seen from the expansion F(z)~

that for given d

IR

1 +C1Z+C,z2+

...

> 0, R cos (J < d and E < R we may choose M so that

-1

In I F (R cos 8 eI'l) I -

£ -1

In I F (E cos 0 et'l) 1/

< MR C0511 O.

Using this, we obtain R

r

-'Ip-(t) dt

•

fA

•

~-Cl(R) II

1 = :be

f

)

(R- 1 1n I F(R cos Oefll ) 1-1- 1 In IF(I cos 8e") ,.

dB +0(1) tost8

and finally

" 1R) z-1I

R

f 'IP~t)

dt

f

=~

R- 1 1n IF (R cos Oet'l) I co:: 8

+ 0 (I).

-~+ .. (B)

o

II

To explain the fundamental role which the class of functions of completely regular growth in an angle plays in uniqueness. theorems, we need a lemma giving a criterion for a function that is holomorphic in an angle to be a function of completely regular growth. LEMMA I. For a function F(z) holomorphic in the angle (at, fJ) and of proximate order p(r) to be of completely regular growth, it is necessary and sufficient that the areal mean

~(R,

a,

~)= 3(p~a)RII

tends to zero as R -...

00.

II

2JI

JJ

(h,(6)-hp,B(6»)rdrdfj

•

B

178

UNIQUENESS, INTERPOLATION AND COMPLETENESS

For a function of completely regular growth we have

PROOF 01- NECFSSITY.

.

Np(r, &, 0)

r+oo

r PIr

11m

[CHAP. IV

)

1

= -SF(n, 27t~3

IJ),

where 8

6)=h~(IJ)-h~(IJ)+p2

sp(a,

f h,(I!()dl!( &

(cf., (3.34) of § 3 of Chapter III). From this equation it follows immediately that

lim

r~:(r) 'f+lc

r+oo

7'

Np(r,

a,

6)d& dfJ

&

8+lc &+1

=

27t!k 3

P"

rr

sp(fl, 6) dfl dlJ

&

_ _1_ { hp(.O+k)-hp(O) - 27tp3 k

hp(&+/)-h p (&)

I

8+lc &+1

9

+ ~; Iff hp(q» df dO dlJ } • &

(4.21)

~

In Chapter lil we proved the following inequality (cf. (3.27» by using the generalized Jensen formula: 8+lc &+1

:k

•

r r

Np(r, n, O)dn d6

&

__I_fr[J~(8+k)-J~(8) _ J~(&+I)-J~(&)] dt k i t

-21t

o

0+,. &+1

+ ~/k J •

0

JJ

hp,r(q»dq> dfl dO.

&

(4.22)

&

As a corollary of Lemma 2 of Chapter III we have for functions of completely regular growth

By using this, we can deduce from (4.21) and (4.22) that '+lc '+1

Jim~/r" r+CID

•

• (h •

r r

p (,)-

hp • r (,) ]d, dfldfJ

= O.

(4.23)

Sec. 2]

179

FUNCTIONS OF FINITE ORDER, HOLOMORPHIC IN AN ANGLE

it follows from the asymptotic inequality

that in (4.23) we may replace the quantity hF.r(O) - hF(O) by its absolute value and obtain the equation 8

lim

f Ihp(!f) -hP.r(IP) IdIP = 0

(0

r .... oo &+1

< 't < l).

Integrating this equation with respect to r from R to 2R we obtain

11m

The necessity is established.

R .... oo

HR,

IJ, ~)

= O.

(4.24)

Let (4.24) hold. If we write

PROOF OF SUFFICIENCY.

II

KF(r) =

JI hp(6) -

hp.r (0) Id'J •

•

then we easily obtain from (4.24) lim R .... oo

~

til

Ii

f

(4.25)

KF(r) dr = O.

Let Ey be the set of values of r for which KF(r»l

Then evidently

r Ii. KF(r)dr>l 1

and by (4.25)

~

> O. ~

mes (E1 ) -

ER mes ( 1)

R

B

m(E~)-m(E:)

_

11m

R .... oo

R

=0.

It follows from this that the relative measure of Ey is zero. Thus if r outside a set El of arbitrarily small upper relative measure, we have

00

IS

lim

r .... oo

rEB"

f IhF(fJ)-hP.r(fJ) IdfJ = O.

(4.26)

&

We now show that the functions IhF(O) - hF.r(O)1 form an equicontinuous family in the angle ex < ex' <; 0 <; {J' < {J for all values of the parameter r, except perhaps for a set El of arbitrarily small upper relative measure (m*(El) < '1/2). To do this we choose a positive number 1 so small that every disk C(R, I) of the form .a'+/l' Iz - Re'-2-1..; IR

180

UNIQUENESS, INTERPOLATION AND COMPLETENESS

[CHAP. IV

IX"

is contained in the region JR ~ Izl ~ tR, arg = <;; p, over which the integral is taken in the definition of 'P(fR, ex, P). It is enough to take 1= t sin ({J - oc)/2. It follows from (4.24) that the average oflh}<"((J) - hF .•(O)1 over the disk C(R, I) tends to zero as R - 00. Hence for sufficiently large R, every disk C(R, /) contains a point zR = R1eiQ1 at which 1 hF (° 1) -

hF. R. (1J 1)

This inequality implies

I

< a.

where N=-( min

hF(fj)-s}.

lI",a<~

We now have the hypotheses of Theorem 7 of Chapter II, and consequently for r E £.Jm*(E2 ) < 1J/2) the family of functions Ihp(O) - hF .•(O)1 is equicontinuous on the interval (oc < oc' " (J " {J' < {J). It follows from equicontinuity and (4.26) that for sufficiently small () > 0, Ikl < (), oc' " 0 " {J', rEEl + E2, we have 8+k

IhF (6) -

hF.,·(O) 1

< ~+ ~

hF(~)- hF,r(~) 1d~

r

1

8

and for r

> r.

°

/hF(fJ)-hF,r(lj) I

< s.

(4.27)

Thus for arbitrary f: > and "I > 0, and for all r not belonging to a certain set of upper relative measure less than 1J, (4.27) holds for oc' " 0 <;; {J'. Then Lemma 1 of Chapter III establishes completely regular growth in every angle oc < (J < {J. The lemma is established. The following lemma also plays an essential role in proving uniqueness theorems by means of (4.20). LEMMA 2. If F(z) is holomorphic and of proximate order p(r)(p > 1) inside an angle larg zl < 'TT /2 + E, E > 0, and of completely regular growth, its zeros satisfy

1 • 'IF (r) p - 1 r-+oo rP(rl

NF (r)

.

- - 11m - - = 11m ----,,....,--:r-+oo rP(rl-l

~

=

" 2'

f.

hF (6) COiP- 2 fJ dlJ,

(4.28)

-"2 where

f r

NF(r) =

G

'IF (I)

- /3 - dl.

(4.29)

Sec. 2]

FUNCTIONS OF FINITE ORDER, HOLOMORPHIC IN AN ANGLE

181

PROOF. In proving this lemma it is natural to use Theorem 3 of Chapter III. Choose IJ > 0 and points 00 < 01 < ... < On (0 0 = -TT/2 + IJ, On = TT/2 - IJ) at which II p(O) is differentiable. Let OJ (j = 0, I, ... , II - I) be the point of the interval [0;, 0Hl] at which cos 0 takes its maximum value. Then we have 71-1

'1p (r)

< ~ n (r cos fJJ, "j,

fJ j +1) +·n

(r sin 3, ;. -

8, ;.)

since the right side is the number of zeros of F(z) in a region containing the y

Fig. 7

disk Iz - r/21 C;;; Chapter III

r/2

(Fig. 7). From this inequality we have by Theorem 3 of

f"

'2

-.-

'Vp(r)

1

11m - - <. -21tp

r~oo rP(r)

cosP 6 dsp (6).

(4.30)

182

[CHAP. IV

UNIQUENESS. INTERPOLATION AND COMPLETENESS

In a similar way we can obtain a lower estimate for the lower limit. For this we let 0; be the point of [OJ, Oj+l] at which cos 0 takes its minimum value, 11-1 obtaining '1p(r)~ ~ n(rcosa;, 6J,6J+1). i=O

From this it follows that

r 1m r-+oo

11-1

( )

r . ..." 1 ~, r P (r) .;v ~ ~

"ip

P

Letting max; 10j+l - Ojl

-+

!fm _ r-+oo

i=O

(SF (aJ+1)

SF (aJ)] COSP 6J

-

•

0 and then !S -+ 0 on the right hand side, we obtain

" 2"

"ip (r) per)

r

~

_1_ 27tp

J

cos' fj dsF(fJ).

" 1"

Combining this with (4.30), we obtain 2"

. "'p(r) 1 hm--=r-+oo rP(r) 27tp

Moreover. if orders,

lim

p

f

cosP adsp (0).

(4.28')

" -"2 # I. (4.29) implies, by means of property (5) of proximate 2"

N p (r)

r-+oo rP(r)-l

.

'" p (r)

=- 11m - - = --p-Ir-+oo rP(r} 27tp(p-l)

f

cosP a dsp(a).

" -2 We now transform the right hand side of this equation. The integral on the right can be written in the form

.

..

"2

2

f. cos

dsp(O)

P fj

=

8

f,. cos I)d [h~ (0) + f hF(!P) d!P! P

p2

-"2

2

"

,.

2

_ f cos

P Ij

dh~(IJ)+ p2

"

f hp(IJ)cosPIj d't. ,.

-2

2

When p

2

> 1 the first integral can be integrated twice by parts "2

" "2

f cosPfjdsp (O)=p(P-I) f "2

We obtain

hp(fj)COSp'-2fJdfj

" -2" and substituting this into (4.28') we obtain the conclusion of the lemma.

(4.31

Sec. 2]

183

FUNCTIONS OF FINITE ORDER, HOLOMORPHIC IN AN ANGLE

We now use (4.20) to prove a uniqueness theorem for functions holomorphic in the half-plane Re z > O. This is an analog of Theorem 1 of the preceding section for entire functions. THEOREM 4. If F(z) (F(O) "" 0) is holomorphic in the right half-plane, of proximate order p(r) (p > 1), and r

NF(r)

=

f - t(t) dt, 'IF

3-

o

then

..

r '2

-.

Np(r)

11m

r-+oo

r

p(r)-1

-<-21 7t

. -'2 ..

(4.32)

hp(fJ)cosp- 2 fJdfJ.

PROOF. Suppose first that hF (?T/2) < 0 and hp( -?T12) < O. Then if 6 > 0 is sufficiently small, the indicator hF (8) remains negative in the angle ?T/2 - 6 ~ 161 ~ ?T12, and for some d > 0 and all r d

>

In IF(re i8)

I< 0

(;

-a-<161-<~).

(4.33)

By (4.20) we have

InIF(rcosOe4'i)1

r P r)

dO cos! 0

+

0(1)

rP (r)-1

and using (4.33) we obtain the asymptotic inequality ~-a

f

2

Np(r) r P (r)-1

1

<~

InIF(rcosOe~)1

rP (r)

dO cos 2 0

-~+5

0(1)

+ rP

ef)- 1 •

2

We transform this inequality by using the fact that for arbitrary sufficiently large r In 1F (r cos 6 ei8) 1< (hp(6)+ IJ (r cos 6)f (r 0088)

(I (j!" i-a) (cf., Chapter I, Theorem 28), and also the fact that

•

11m r-+ao

(r

cos O)P L (r cos 0) rPL (r)

=cos

P" \I

6

and all

184

UNIQUENESS, INTERPOLA nON AND COMPLETENESS

[CHAP. IV

<

uniformly for 161 ."./2 - lJ. After making the transformation we obtain the asymptotic inequality ~-a 1

N,(r)

f

1

r,(r)-l

< 21t

,-1OdO+

(h,(fJ)+a) cos

0(1) r P (r)-1

-~+a

and since

E

> 0 is arbitrary

2

~-a

f 2

N,(r) 1 11m p (r/- 1 "" "r-+co r ......

P 1

h,(O) cos - 0 d8.

-~+a

Finally, since

{J

2

> 0 is arbitrary _.

N,(r)

11m

r'

r-i'CO

•

2'

)

(r/-l " " -

~

r

. h,(O)cosP-

2

0 dlJ.

~

•

We now dispose of the other possibilities h F l-."./2) < 0 and hF (7T12) < O. To do this we construct an entire function (z) of proximate order p(r) which is of completely regular growth and has an indicator hlb(8) such that hlb{."./2) < -h p (7T/2) and h lb( -7T/2) < -h1<'( -7T/2). The possibility of such a construction follows from Theorem 3 of Chapter 11 when p > 1. The function 'If (z) = P (z) ell (z) has the indicator h", ( - ;) 0 H h", (; ) 0

<

<

and consequently h'l.( -7T/2) < 0 and h'f"{7T/2) < O. We now have the inequality • 2' N", (r) I lim , (r/-l " " "h", (0) C05,-2 0 dfJ, r-+co r ......

f •

which can be written in the form -

N,(r)

11m () r-+co r P r -

1

.NIbCr) + r-+co lim (-) rP r -

-1

•

~

"" ~ J•

hF(fJ)COsP-ZfjdO+

The limit

•

~

2

f•

-.

h.(6)cosP-ZOd/J.

Sec. 2]

FUNCTIONS OF FINITE ORDER, HOLOMORPHIC IN AN ANGLE

185

exists by Lemma 2 and is equal to the second term on the right. Hence we obtain

The theorem is established. In the preceding section we derived from Theorem 1 an estimate for the upper density A of the set of zeros of an entire function. In an analogous way we can derive from Theorem 4 an estimate for the quantity ~1'(r)

-

Um --p(r)'

.,.-+CD

Under the hypotheses of Theorem 4

COROLLARY.

For k

'"

r

_~_1'_~_) ~ 1_+---!.:~)_P_

-Um ro+CD

PROOF.

r

r P( J

---=.,( ~ (p -1)

cos PfJ

~1'(8).

(4.34)

1/

'" -2'

> 1 it follows from the monotonicity of "~r) that k,"

J'F(r) ( 1) -r- 1 - k

F -< J' -t-dt -< NF(kr). '1

(t)

2

t"

Hence by using (4.32) and (4.31) we obtain To

lim

~F(~) ~

r-+,,? r P ( I

"2

kP

~(p-l)p(k-l)

fcos p /jdSF(fJ). r.

2

The right hand side of this inequality takes its smallest value for k = 1 + 1/p. Putting in this value of k, we obtain (4.34). Just as in § 1, we can construct a funct:"n for which there is equality in (4.34); but we shall not go into this. An estimate for the lower limit

leads us to a theorem analogous to Theorem 2 of the preceding section. THEORllM 5. If F(z) is a holomorphic function, not identically zero, in the half-plane Re z :> 0, of proximate order per) with p > I, then

'" "2

"p(r) 1 lim ----p\r) ~ r 2r.p

r -+ CD

f '"

cosP fJ dsp(Ij).

(4.35)

188

UNIQUENESS, INTERPOLATION AND COMPLETENESS

[CHAP. IV

We now apply Lemma I. To do this, we multiply the integral on the left of the last equation by rlR2 and integrate from R to 2R sec y. We obtain 11m _1 8-+00 RI

J' JI

18880

1

In IF (r cos 6e4') I rP Ir)

-1

Denoting r cos 0 by r1, we have 2B cos8 T

lim _1 84>00

RI

J'

j

c08T

-T

and finally

81~moo ~

lin IF (rle") I rP Ira' 1

1J hF.r(~)-hF(fj)lrdrdfj T 28

I

= O.

From this it follows by Lemma I that F(z) is of completely regular growth in the half-plane Re z ~ O. The theorem is established. As a corollary of this theorem we obtain the following fundamental theorem of M. L. Cartwright [3]. THEOREM 6. If F(z) is holomorphic and of proximate order p{r) in the angle (Qt, p) and has a sinusoidal indicator for Qt < 0 < p, and the opening of the angle (Qt, P) is greater than 171 p, then it is of completely regUlar growth and the density of its zeros in the angle is zero. PROOF. Without loss of generality we may suppose that the angle (Qt, P) is the half-plane Re z > 0 and that p > I. For a sinusoidal indicator, ds F «() = 0, and hence there is equality in (4.35). It follows from Theorem 5 that F(z) is of completely regular growth. In addition, the density of its set of zeros in the half-plane Re z > 0 is 1/217p[sF(17/2) - s/<.( -17/2)] = O. The theorem is established. It is interesting to compare this theorem with Theorem 5 of § 3 of Chapter III. That theorem requires that the indicator be sinusoidal and that the angle contains a rayon which F(z) is of completely regular growth. It follows from Cartwright's theorem that if the opening of the angle exceeds 171 p, the requirement of the existence of such Ii ray is superfluous. All the theorems that we have proved on estimates for the density of the set of zeros in a half-plane concern the case p > I. The case p = I is exceptional. In this case there is no estimate of the kind given in Theorem 4. We consider only functions of exponential type and prove a theorem which contains, in particular, the theorem of Cartwright that we stated at the beginning of this chapter.

Sec. 2]

FUNCTIONS OF FINITE ORDER, HOLOMORPHIC IN AN ANGLE

189

THEOREM 7. If F(z) is h%morphic, not identically zero, and of exponential 0, then type in the half-plane Re z

>

(4.36)

PROOF.

We stut from formula (4.20). For a given

JnIF(re'~)1 we can choose d

B

> 0 such that

(101 <.;)

< IhF(rj)+ajr

> 0 so that

" -"2+«(r) (a(r) = arcsin

f).

We shall show that the first term on the right is the principal term. To do this, integrate by parts. We obtain

'"

"2- II (r)

NF(r) <

I

2~ IhF(IJ)+ejJnltg(~+-i)1 "

-2"+II(r)

'2" II (r)

f I

In tg

"

(~ + i-)Ih~ (IJ) db = 2~ [hF (~._- ex (r»)

-"2+11(r)

Since In ctg

IX;) / In r -+ 1 as r -+ 00, it follows from this that Np(r) <

and since

B

~

[h p (;)+h p (-i)+2e]lnr+O(lnr),

> 0 is arbitrary -

Np (r)

1 [

( 1t )

(11: )]

rl~,: Irir < 2it h p 2" +hp - 2" • We turn now to the quantities

N~r)

and

"~r).

'1p(r)

1'= lim - - co r-+

r

Put

(4.37)

190

UNIQUENESS, INTERPOLATION AND COMPLETENESS

Then for every

E

[CHAP. IV

> 0 we have the asymptotic inequality "f(r) > (:J.--e)r.

Using this, we find from the definition of N ]<.(r) that

> 0 (l)+ (p. -

N]<"(r)

Finally, since

E

e) In r.

> 0 is arbitrary this implies ~

IJ.< I

---.-

. NF(r) 11m - 1 nr- '

__

1" .... O'J

Substituting this into (4.37), we obtain

JlF(r) I [ lim - r - ~. 2r. hF

(Tt2" ) +hF (r.:)] -2" .

r-+oo

The theorem is established. We note in addition that the width h1<"(17/2) hF( -17/2) of the indicator diagram can be represented in the form

+

r.

hF(~) + hF(- ~) =

"2

J

cos 0 dSF('J)·

" -"2 In fact, it is easily seen that the integral on the right side of this equation gives the projection on the imaginary axis of the arc of the boundary of the indicator diagram of F(z) between the supporting lines '-,,/2 and 1"/2 and consequently is equal to the width of the indicator diagram in the direction of the imaginary axis. Therefore (4.36) follows from (4.35) if we put p = 1.6 3. Functions Vanishing on a Set with an Angular Density7 In this section we give a different generalization of Carlson's theorem for holomorphic functions of finite order in an angle. We consider functions which vanish on a set of points that has an angular density, and show that under certain restrictions on the growth of the function, the function must be identically zero. THEOREM 8. Let F(z) be holomorphic and at most o/normal type with respect to the proximate order p(r) in the angle ex ..;;; arg z <: ex + 17/ p, and vanish on a set N = {ak } in this angle, with angular density ~.y{tp), and let

-+-P"

f cosp(la-'~I-1t)d~N('!.)

HN(fJ)=_1t_ sin 1tp •

..

T

• We note that in this case it is an open question whether there is equality in (4.36) only for functions of completely regular growth. , B. Ja. Levin [IJ.

Sec. 3]

FUNCTIONS VANISHING ON A SET WITH AN ANGULAR DENSITY

191

when p is non integral and ~

'It

J

P

sin p I'?-- 0 Idj.N ('?)

a

when p is integral. Then if F(z) is not identically zero,

PROOF.

Consider the holomorphic function lJI'{z) =

in the angle (ex, ex

+ 1T/ p), where
F(z) «I>N(Z)

IT O(a:; p) k=l

when p is not integral and
n

= -(

I - 2)

k=l

Z2p\

ai

when p is integral. Since (z) is a function of completely regular growth with indicator H N(() (cf., Chapter II, § 4) we have h,l' (lJ) = hF(~) - hN(fj)· By property (h) of the indicator (cf., Chapter I, § 16)

and consequently hp (a

+~) +hp(a) >.Ii N( a+

-}} +H.v(a).

The theorem is established. We note that whe i p = 1 the quantity h~ex + 1T) + hF(ex) is the width of the indicator diagrar. of the function in the direction arg z = ex. If p(r) = I and the set N is the se~uence of integers al: = I, 2, ... , then H,.... (O) = 1T Isin 01, and by the theorem just established

hp(i) +hp( - i) ~ 27r. Hence Carlson's theorem is a special case of Theorem 8. We obtain a different uniqueness theorem by using property (g) of the indicator instead of property (h). Before stating this theorem we introduce the

192

UNIQUENESS, INTERPOLATION AND COMPLETENESS

[CHAP. IV

concept of an indicator of uniqueness. We call a trigonometrically convex function h(O) the indicator of uniqueness of the regular set N with angular density d.v(O) and indicator H.v(O) if there is no entire function F(z), not identically zero, of proximate order p(r) with indicator h(O), which vanishes at all points of N. THEOREM 9. In order that a trigonometrically convex function h(O) should not be the indicator of uniqueness of a gil'en regular set N, it is necessary and sufficient that the following inequality holds for arbitrary ex and fJ (ex < fJ):

sea ± 0,/3 ± 0) > 271'p(AN(~-+-0)-~N(IJ±O)J.

(4.38)

where 5(a.

~)=h'm-h'(a)+p2

II

Jh(tp)d'P. «

We note first that h(O) fails to be an indIcator of uniqueness if and only if it can be represented in the form PROOF.

h (0) = H N (6)

+h

J

(4.39)

(r ).

where h1(O) is a trigonometrically convex function of period 217'. In fact. if h(O) is not an indicator of uniqueness, there is an entire function F(z) of proximate order p(r) with indicator h(O), vanishing on N. Let ".(z) be the canonical function of the regular set (Chapter n, § 1). Then evidently the function

'Hz) = is entire and its indicator

F(z) WN(z)

hlF ('J) = h (II) - H N (II).

Conversely, if (4.39) is satisfied then, constructing an entire function 'fez) with indicator h1(O) (Chapter II, § 4), we see that the function F (z) = rj'N (z) lr (z) has the indicator h ,/;I) = HN (11)+ hi (It) and vanishes on N. To establish the equivalence of (4.39) and (4.38) for arbitrary IX and fJ (ex < fJ) it is enough to show that it is equivalent for a trigonometrically convex function h1(8) to satisfy the inequality

Iz~ (~) - h~ (a) + r''J for ex

~

f

hI (II) db ::p 0

< {J (Chapter I~ § 16) or the equauoh H~O)-H~(a)+p9

~

f HN(q» •

established in § 3 of Chapter III.

tkp

= 271'p[AN(P)

- 4 N(a)],

Sec. 3]

FUNCTIONS VANISHING ON A SET WITH AN ANGULAR DENSITY

193

We recall that for p = 1 the quantity s(ot, p) is the length of the boundary of the region' of which h(O) is the supporting function, between the points of contact of the supporting lines perpendicular to the rays arg z = ot, argz = (J. Hence for p = 1 we have a geometrical criterion for the supporting function h(O) of a convex region to be an indicator of uniqueness. In fact, the length of the arc in question must be less than 211' times the density of the set N in the angle. We also remark that the function HN(O) minimizes the functional ~

s(a,

p,

h)= h'O)-h'(cx)+pi

f h(lJ)d6,

..

with arbitrary ot and {J (ot < (J), among all trigonometrically convex functions that are not indicators of uniqueness for a given regular set. In fact, s(cx, ~, HN)=21tp[AO)-A(cx»), and in the general case it follows from (4.39) that s(cx,~,

h)=21tp[£\(p)-a(cx»)+ s(a,p,h).

But by property (g) of trigonometrically convex functions (Chapter I, § 16), s( ot, (J, hI) ;;> 0 and consequently

s(cx,

~,

h):;ps(cx,

~,

H N ).

We note (Chapter I, § 16) that the equation s(ot, (J, hI) = 0 is possible only when hl(O) = A cos pO + B sin pO. It follows from this that when p is not integral H N(O) is the unique function minimizing the functional s(O, 211', h), and when p is integral the general form of minimizing functions is h(O) = H.y{O) + A cos pO + B sin pO. In an analogous way we can define an indicator of uniqueness for holomorphic functions of proximate order per) in an angle (IX, (J). We shall call a trigonometrically convex function h(O) (IX <; 0 <; (J) an indicator of uniqueness for the set N which is inside the angle (ot, (J) and has angular density a~O) provided that there is no function that is holomorphic in that angle, of proximate order per), with indicator h(O), vanishing at all points of N and not identically zero. We can prove the following theorem about indicators of uniqueness for an angle. THEOREM 10. If the trigonometrically convex function h(O) Cot <; 0 <; (J) is not an indicator of uniqueness for the set N, then the following inequality holds for arbitrary ot' and {J' (ot <; ot' < {J' <; (J):

(4.40)

194

UNIQUENESS, INTERPOLATION AND COMPLETENESS

PROOF.

[CHAP. IV

Suppose first that p is not integral, and construct the function 00

til (z) =

IT a (~; p), k=l

ale

where ak are the points of N that are in the angle (ex', (l'). Let F(z) be a holomorphic function of proximate order p(r) in (ex, P) with indicator h(O). Then 'I'(z) =

~~~

is a holomorphic function in (ex', P') of proximate order p(r), with indicator hll"(6)= h(6)-HN(fJ). It follows that s(<<·.

P',

h)-s(<<'.

W,

HN)~

O.

and since ct>(z) is of completely regular growth, we have s(<<', ~/,HN)=2'itp[aN(p')-aN(a')].

To treat the case of integral p, we first note that it is enough to prove (4.40) when P' - ex' < 7T/p. In fae:t. if we have proved this, then to prove (4.40) in general we have only to divide (ex', P') into parts and add the corresponding inequalities. Put

~ (z) --;-

IT (I - ::: ), k=1

k

where ak are the points of N lying in the angle (ex', P') (P' - rx.' < 1T/p). The function ct>(z) is holomorphic, of proximate order p(r), and of completely regular growth. The rest of the proof is just the same as for nonintegral p. 4. Representation of Entire FUDctioDS by Lagrange Interpolation Series

A problem closely related to that of the unique determination of an entire function by its values on a set of points is the problem of interpolation of entire functions, i.e., the problem of constructing an entire function that belongs to a prescribed class and takes given values {b .. } at the points of a given set {a .. } (nodes). For this constCl!ction we use interpolation formulas which generalize the corresponding formulas for interpolation by means of polynomials. A Newton series is an interpolation series of the form 00

~ an (z 11-0

Gr.1)(z -

Gr il)

.••

(z ~ Gr'I)'

Sec. 4]

ENTIRE FUNCTIONS BY LAGRANGE INTERPOLATION SERIES

195

It is known 8 that when an = n (n = 0, 1,2, ... ) the region of convergence of this series is a half-plane Re z ;) r (y is a real number). In addition, the sum of a Newton series is a function of exponential type in this half-plane and its indicator satisfies the inequality

hF(~)< cosfJ In(2cosfj) +ll sinO

(101 < ;-).

On the other hand, by Carlson's theorem a function of exponential type in the half-plane Re z ;) y is determined by its values at the integral points n = 0, 1, ... if hF(1T/2) + hF( -1T/2) < 21T. Hence there is a class of functions of exponential type which are determined by their values at the points 0, 1, 2, ... but cannot be constructed from these values by means of a Newton series. s In particular, this class contains all functions of exponential type for which 1T < h F(1T/2) + hJo'( -1T/2) < 21T. Hence for the set 0, I, 2, ... the "convergence class" is narrower than the "uniqueness class" defined by hF(1TJ2) + h F( -1TJ2) < 21T. In many cases it is more convenient to represent entire functions by a different interpolation formula, which represents an entire function, not as a series of polynomials, but as a series of special entire functions. Let fez) be an entire function which has simple zeros at all the nodes {an} (Ianl- 00), and let {b n } be values assigned at the nodes. If the series co

~

bnl(z) ~ I' (an) (z - an)

(4.41)

"=1

converges uniformly in every bounded region, it represents an entire function which solves the interpolation problem. This series is called a Lagrange

interpolation series. We shall investigate the question of the representation of entire functions by Lagrange interpolation series in the case when the nodes form an R-set with index 10 per) (for the definition see Chapter II, § I). We denote the set of nodes by 21, the canonical functionf(z, 21) of this set by fez), and the indicator H'II.(O) of the set by H(O). The following theorem about the convergence of the Lagrange interpolation series then holds. THEOREM II. If the set 21 of nodes is an R-set (Chapter II, § 1) with index p(r) and iffor some e > and n > nee) all the numbers bn satisfy

°

In Ibill

< IH('~n)- 311 an 1P(I...I)

(9n = arga n ),

(4.42)

• Cf., A. O. Gel'fond [1; 1]. The question of the representability of entire functions by Newton series has been investigated in a number of papers. The most complete results are obtained in the papers by A. O. Gel'fond [1] and I. I. Ibragimov and M. V. Keldy~ [1]. Gel'fond gave a very general method for constructing an entire function from its values on a sequence of functionals [1; 1]. 10 Part of the results of § § 4 and S are given in the author's paper [1]. See also A. Pfluger [1]. I

196

UNIQUENESS, INTERPOLATION AND COMPl.ETENESS

[CHAP. IV

then the Lagrange series (4.41) converges uniformly in cury bounded region and the limit is an entire jUllctioll lI,'irh indicator nut exceeding H«(). PROOF.

By Theorem 5 of Chapter II,f(z) satisfies the asymptotic equation In If (re i8 ) I ~ H ('J) rP (rl

everywhere outside the excluded C R-disks. We now estimate the function In 1 I(z)

z-a"

1[,(an)l.

Consider

'I.

It is harmonic outside the excluded C R-disk with center at a .. and satisfies the foIlowing asymptotic equation (as n - 00) on the boundary of this disk: In 1 I(z) I ~ H (.~ .. ) I aIlIP(lan " •

z-a"

By the maximum and minimum principle for har ••lOnic functions we obtain (4.43)

From this and (4.42) it follows that when z lies outside the excluded eR-disks, and n > n'(e), bra

In 1 I' (a .. )(z - an)

I<-.!.\a I p(/a,," . 2'1

On the other hand, we have and consequently

l aII I Puanl)--- ~_n_. (21t) row

This relationship makes it possible to write the preceding inequality in the form

II'

bn

(a,.) (z - an)

1<

,-Crt

(4.44)

Hence the series

converges outside the excluded CR-disks to a function q:>(z). Applying the maximum principle in each excluded disk to the sections of the series (4.41) of entire functions, we find that the series converges uniformly in each bounded region to an entire function F(z) = j(z)cp{z). We show further that h,(6) <: H(fJ). To do this we first show that cp(z)

Sec. 4]

ENTIRE FUNCTIONS BY LAGRANGE INTERPOLATION SERiEs

tends to zero as Izl for z E Cn

-+ 00

outside the set of CR-disks. In fact, we have from (4.44) Ibnl

N

I ~ (Z) I < ~ II' (an) II z -

,-eN

an I

"=1

For any

E

197

+ 1 _ ,-e·

> 0 we can choose N, and then r" so that I!p(z) I < E

for Izl > r. and z E C•. Therefore

outside the excluded disks The theorem is established.

I F(z) / < I/(z)/ and for Izl > r.. and

consequently hF «()

< H«().

REMARK I. The function F(z) represented by the series (4.41) is not the unique solution of the interpolation problem el'en under the condition

hp (6) ~ H (6).

(4.45)

We shall show that all solutions that satisfy this condition are represented by the formula (4.46) ~(z)= F(z)+(I)(z)/(z), where w(z) is an entire function of at most minimal type of the proximate order per). In fact, if (z) is any solution, then (I)(z)

= ~ (z) -F(z) fez)

is an entire function. In addition, if (z) satisfies (4.45) the indicator of the numerator does not exceed H«(). The denominator is an entire function of completely regular growth with indicator H«(). Hence the indicator of w(z) is at most zero and consequently w(z) is at most of minimal type ofthe proximate order p(r). Conversely, every function (z) represented by (4.46) is evidently a solution of the interpolation problem under the supplementary condition (4.45). REMARK 2. Ifwe replace the supplementary condition (4.45) by the stronger condition Ii I ~(Z)I - 0 (4.47) IH~l_ I/(z) I - , .("0

which, as has been shown, F(z) satisfies, we find that the entire function w(z) tends to zero when Izl -+ 00 (z E C), and by the maximum principle it follows that w(z) ~ O.

198

UNIQUENESS, INTERPOLATION AND COMPLETENESS

[CHAP. IV

Hence F(z) is the only solution of the problem under the supplementary condition (4.47). In the following theorem we give a sufficient condition for the representability of a given entire function by a Lagrange interpolation series. THEOREM 12. If the points {a k } form an R-set entire function F(z) that satisfies hF(6)

< H(fJ)

(0

m:

with index p(r), then every

-< 6 < 21t)

(4.48)

is represented by the series 00

F(z)= ~ /(an)/(z) ~

n=t

f

,

(an) (z - an)

(4.49)

with uniform convergence in each bounded region.

PROOF. We shall show that when (4.48) is satisfied the sequence {b n = F(a n )} satisfies (4.42) of Theorem 11. In fact, it follows from (4.48) that for some

e>O (4.50)

In addition, by Theorem 28 of Chapter I we have the asymptotic inequality

IF (re i8 ) 1< [h F (fJ)+ e) rP(r) < [H(O) -e] rP(r) and consequently for n > nee) In I F (a,,) 1< [H (Ill,,) - e) I a" IP(lan ". In

By Theorem 11 this inequality implies the convergence of (4.49). We still have to show that the sum of this series is equal to the given function F(z). To do this we use Remark 2 after Theorem 11. As we see from (4.48), F(z) satisfies (4.47). The sum of the series satisfies the same condition. Then (4.49)}01l0ws from the uniqueness of the solution of the interpolation problem under the supplementary condition (4.47). The theorem is established. REMARK. Theorem 12 remains true if the requirement that the set of nodes is an R-set is replaced by the much weaker requirement that it is a regular set. In this case, however, the sum of the series must be taken in a special sense. Indeed, we can say that •

~

REE

I an I
F(z)= R-+oo hm

F(~)f(z) I'( an)(z _ an )'

(4.51)

where E is an EO-set. In fact, from (4.48) and from the asymptotic equation Inlf(rei~1 ~ H(8)rP(r), which holds for r not belonging to an EO-set E, it follows that

Inl I

F(r"") (r")

1<

_arP(r)

(rEE).

Sec. 4]

ENTIRE FUNCTIONS BY LAGRANGE INTERPOLATION SfRIES

i

199

This inequality implies that the integral ) -

lr (z -

1

27t1 I~

=,.

F(C)/(z)

I (C) (C _ z) d:

tends to zero when r tends to infinity outside the set E. On the other hand, ~

~

lr(z) = F (z)-

F(an)/(z)

I' (an) (z - an)

Ian 1<"

and consequently (4.51) holds. We call a trigonometrically convex function h(O) a cOnL'ergence indicator (for a given p(r)) if every entire function of proximate order p(r) with indicator not exceeding h(O) is represented by the series (4.49). It is clear that a convergence indicator is also a uniqueness ir,dicator. It follows from Theorem 12 that a convergence indicator can be as close as we please to a set indicator H(O) which is not a uniqueness indicator. We recall that the situation is different for the Newton interpolation series. l1 It was shown in Theorem 11 that if an R-<;et fan} is giver and the sequence {b n } satisfies (4.42), there exists an entire function F{z) wp;)se indicator does qot exceed the set indicator H(O) and which takes the values {an} at the points

{b n J. Tn the following theorem we give a necessary and sufficient condition for a given sequence of numbers {b n } to be the values, at the points {a,,} of a given R-set, of an entire function whose indicator does not exceed the set indicator

H(O). THEOREM 13. Let III be an R-set of points {an} with index p(r) and {b"} a sequence of numbers. A necessary and sufficient condition for the existence of an entire function F(z) satisfying the conditions

F(a n ) = bn ,

(a)

hp (li)

(/1)

is that

-. { lnlb,,1 hm (J I) la"IP a.

- H.(I},,)

-< H(6),

} <0

(4.52)

"--+00

PROOF.

inequality

The necessity of (4.52) follows immediately from the asymptotic In I F (re,a) 1< (hp(Ii)+ s) rp(rl,

which the interpolating function F(z) must satisfy, and (fJ). l\ We may, however, remark that the solutions of many problems of the theory of entire functions require the expansion of a function io a Newton series rather than a Lagrange series. This is basically connected with the fact that a Newton series is a series of polynomials.

198

UNIQUENESS, INTERPOLATION AND COMPLETENESS

[CHAP. IV

Hence F(z) is the only solution of the problem under the supplementary condition (4.47). In the following theorem we give a sufficient condition for the representability of a given entire function by a Lagrange interpolation series. THEOREM 12. If the points {a k } form an R-set entire function F(z) that satisfies hp(6)

< H(fJ)

(0 <. 6

mwith index per), then every

< 2n)

(4.48)

is represented by the series 00

F (z)

=

~

~

n=l

f

/

(an) f (z) , (an) (z - an)

(4.49)

with uniform convergence in each bounded region.

PROOF. We shall show that when (4.48) is satisfied the sequence {b n = F(a n )} satisfies (4.42) of Theorem 11. In fact, it follows from (4.48) that for some

E>O (4.50) In addition, by Theorem 28 of Chapter I we have the asymptotic inequality

In I F (re i9 )

i < [h F (0) + e) rP(r) < [H (0) -

> nee) In I F (an) I < [H (7n) -

e) rP (r)

and consequently for n

e) I an IP(/ an

I).

By Theorem II this inequality implies the convergence of (4.49). We still have to show that the sum of this series is equal to the given function F(z). To do this we use Remark 2 after Theorem 11. As we see from (4.48), F(z) satisfies (4.47). The sum of the series satisfies the same condition. Then (4.49)Jollows from the uniqueness of the solution of the interpolation problem under the supplementary condition (4.47). The theorem is established. REMARK. Theorem 12 remains true if the requirement that the set of nodes is an R-set is replaced by the much weaker requirement that it is a regular set. In this case, however, the sum of the series must be taken in a special sense. Indeed, we can say that (4.51)

where E is an EO-set. In fact, from (4.48) and from the asymptotic equation Inlf(rei~1 ~ H«()rp
IIF(r~) 1< (rea)

arP(r)

(r EE).

Sec. 4]

ENTIRE FUNCTIONS BY LAGRANGE INTERPOLATION SF;RIES

I

199

This inequality implies that the integral -

fr (z) -

1 27t1

F(C)/(z) z)

I (C) (C _

d:

I~ =r

tends to zero when r tends to infinity outside the set E. On the other hand, fr (z)

= F (z) -

~

F(an)/(z) an)

~

I' (an) (z _

Ian I
and consequently (4.51) holds. We call a trigonometrically convex function heel a cOnL'ergence indicator (for a given per)) if every entire function of proximate order per) with indicator not exceeding h(O) is represented by the series (4.49). It is clear that a convergence indicator is also a uniqueness ir.dicator. It follows from Theorem ! 2 that a convergence indicator can be as close as we please to a set indicator H(e) which is not a uniqueness indicator. We recall that the situation is different for the Newton interpolation series. l1 It was shown in Theorem II that if an R-c;et fan} is giver and the sequence {b n } satisfies (4.42), there exists an entire function F(z) wpose indicator does I!0t exceed the set indicator H(e) and which takes the values {an} at the points

{b n }.

rn the following theorem we give a necessary and sufficient condition for a given sequence of numbers {b n} to be the values, at the points {an} ofa given R-set, of an entire function whose indicator does not exceed the set indicator H(e). THEOREM 13. Let ~ be an R-set of points {an} with index per) and {b n } a sequence of numbers. A necessary and sufficient condition for the existence of an entire function F(z) satisfying the conditions

F(a n )

=

bn ,

(a)

hp(6)

<: H(6),

(tJ)

is that

(4.52) PROOF.

inequality

The necessity of (4.52) follows immediately from the asymptotic InIF(re,Q)1

< [hp(6)+Ii) rP

(r) ,

which the interpolating function F(z) must satisfy, and (fJ). II We may, however, remark that the solutions of many problems of the theory of entire functions require the expansion of a function in a Newton series rather than a Lagrange series. This is basically connected with the fact that a Newton series is a series of polynomials.

200

UNIQUENESS, INTERPOLATION AND COMPLETENESS

[CHAP. IV

To establish the sufficiency we note first that if

-I"

{ln1b l n

n~ lanlP(la..l) -

H()}

0 <,

tltn

the existence of an interpolating function satisfying (fJ) follows dilt:dly from Theorem II. However, if there is equality in (4.52), the Lagrange series (4.41) does not necessarily converge, and hence we cannot construct the function F(z) in the same way as in Theorem II. To obtain a convergent series in this case, we adjoin new nodes to the set '}! in such a way that the new set is an R-set and the canonical function h(z) of the new set grows faster than f(z). When h(z) grows sufficiently rapidly the series

converges, and the interpolating function F(z) can I". constructed with the aid of the Lagrange formula. At the additional noel:.:" :X:n we may put F(:x: n ) = O. We also require that the indicator hjo'(O) of the function we have constructed does not exceed H(O). This will be true if we extend the set'll in such a way that its indicator is not changed. We make this extension in the following way. When there is equality in (4.52) we have the equation

By Theorem 16 of Chapter I there is a proximate order p(r) such that12

11m (lan./-Pt(IGnl)/n!

n+oo

G

and lim

/n

f~~

I) = J,

r P' (r)-p (r) = O.

r+oo

We introduce the proximate order

(r»)

p P~ (r) = max ( PI (r), -2-

and construct an R-set '1(1 with index P2(r} satisfying condition (C') (Chapter IT, § l) and such that the indicator H[!.)O) satisfies

H •. {fJ}

>2

(0

<. 6 < 21t).

II In Chapter I we considered not sequences but functions defined on the positive real axis; however, in the present problem the difference is not important.

Sec. 4]

ENTIRE FUNCTIONS BY LAGRANGE INTERPOLATION SERIES

201

Here we may suppose that the excluded disks for the sets III and ~l do not intersect. 13 Let w(z) be the canonical function of ~l. Outside the excluded disks for ~l it satisfies the asymptotic inequality In I w(z)

I > 2r P1 (r)

and consequently In

It follows

IIf' (an)

bn CIl

(an)

1< - (I-e)/a

n

IPa(lan ".

l' from this inequality that the series 00

f (z) CIl (z) bn ( ) - ~ f' (an) '" (an) (z -

F z _ '\'

an)

n=1

converges uniformly in every finite region and solves the interpolation problem, and the indicator of the function represented by this series does not exceed (H'!I().

11 We outline the construction of 'HI: on the rays arg z = 2.,k/n + IX (k = 0,1,2, ... , n - 1; n integral and n > 2p.; oc arbitrary) we select the points where the function Ilr P.(r) takes integral values. The set so constructed has an angular density. When p. is integral we take n to be the even number 2P2 and condition (8) (Chapter II, § I) is satisfied. In addition, for the points on a single ray we have by construction 1 = III an +lI Pa ( I an + 1 1 ) -Ill II" IP1l
= Il ( 14n+l I-I lin 1) [Pa (c) cPo (C)-1 + cP1 (C)P2(C) In c) (Ian I
or asymptotically

I an+tl-I an I>

i

1l- 1p2"1 I an 11 - p, II an P,

and consequently (C') is satisfied. Using the asymptotic formulas (2.05) and (2.07), we find that HI(O) > 2 for sufficiently large fl. We also need to know that the excluded disks for 'H and 'H' do not intersect. Note that the diameters of the excluded disks for 'HI do not eltceed P ( IGn Il

dlanll-Pallanll
2

(*)

If 'H satisfies (C') it is enough to take the rays containing points of II, different from the rays containing points of Ill, and choose d sufficiently small; then the eltcluded disks of the two sets cannot intersect. If, however, III satisfies (C), the rays containing the points of 'H, can be chosen arbitrarily. If some point of 'H, falls inside an excluded disk Cw, we may move this point to the circumference of the disk. After this transformation Ill, is still an R-set and its indicator is not changed. If we halve the radii of the excluded disks, then, as we see from (.), the disks for III and
202

UNIQUENESS, INTERPOLATION AND COMPLETENESS REMARK.

[CHAP. IV

The hypothesis that the set of nodes is an R-set can be weakened to .

{ln1f'(a,,)1

hm

lanlP(I all I)

ft--+OO

- H(t/I,,)

}

= O.

We shall not discuss the proof of this. 144 5. Some Applications of Lagrange Interpolation Series By using Lagrange interpolation series we can investigate the question of how the behavior of an entire function in the whole plane depends on its behavior on a set of points with infinity as a limit. We shall discuss several results from this range of ideas. THEOREM 14. If the entire function <J>(z) is (I) bounded on an R-set K'ith index per) and posith'e indicator (H(O) > 0 for 0 OS;;; 0 < 27T), (2) of at most normal type of the proximate order per), and its indicator satisfies the inequality

(j=1,2, ... ,n),

h
where

10j ~! then <J>(z)

OJ I (:: ~ ,

I /j 1 -

rj n

(4.53)

+ 2'1t I -< ~

== const.

We show first that the hypothesis of the theorem implies h~(/J) H(rJ) (0 6 2'11:). (4.54) To do this we construct a function 'I'(z) of completely regular growth of proximate order per), for which the indicator h't'(O) is a constant h less than PROOF.

-< <

<

min (H(6 j ) -

h~ (/jj)]

(j = I, 2, ... , n),

and

h

<

min H(/J). 0.;;6<211:

The indicator of <J>(z)'I'(z) is equal to the sum hlll(O) + h and consequently satisfies the condition (4.53). Using this, we shall show that <J>(z)'I'(z) can be represented by a Lagrange interpolation series. In fact, since the numbers <J>(aJ are bounded and h't'(O) = h, it follows that 1-;- In I ell (a,,) \11' (a,,) I / ' I 1m

n-+oo

(I

hI QIII P

I

an)

<:::,..

14& Firsakov [1] proved the following stronger theorem: Let the set of interpoloJion nodes {~} be regularly distributed with respect to the proximllle onIer p(r) and have indicator H(fI). In on/er thol to a $equence {a,.} satisfying

lim {

"......,

Inla" I

fA..lp<1Ao1>

-H(",,)} <0

"

(",,,_arg~>,

there corresponds at/east one entire junction, with indicator ".(fI) tion problem, it is nece&Stuy and sufficient thol

lim

".......

{lnlrhl + H(",,,>} < 1A..11'(1Ao1)

where .f(z) is the canonical junction oj the set

{A..}.

< H(fI), t1lm solves the iIIIe1pOiD-

0.

Sec. 5] Because h

APPLICATIONS OF LAGRANGE INTERPOLATION SERIES

203

< min o<9<2" H(O) we obtain -I' In I ~ (an) IV (an) 1m

ft-+oo

H (4'n)

(I

I an IP

an

I

Il

< 1'

and by Theorem II the series

converges and, moreover, as has already been shown, tends to zero as Izl ->- 00 with z outside the excluded set of CICdisks. The entire function O(z)f(z) take:" the values <1>(a n)\f"(a,.) at the points {an}. Hence the difference

C(z) = -II (z) 'F (z) -

ij

(z) / (z)

is an entire function vanishing on the pomts {an} and consequently ~ (z) = ~ (z) IV (z) _ 0 ( ) ( ) = f(z) Z Z f(z) Z is also an entire function. It is easily seen that X(z) is at most of normal type of the proximate order p(r). We shall show that X(z) == O. To do this, we remark that the indicator h(O) + h of the function <1>(z)o/(z) is less than H(O) at the points OJ (j = 1,2, ... ,n), and by continuity we have the inequality.

h,z,(fJ)+h

< H(6)

inside the angles 10 - 0;1 ~ ~ (j = I, 2, ... ,n) for sufficiently small () > O. From this it follows that inside the angles 10 - 0;1 ~ b/2 (j = 1,2, ... ,n), but outside the excluded C]Cdisks, the function I}>

(z) IV (z) f(z)

tends uniformly to zero as Izl ->- 00. Since O(z) also tends uniformly to zero outside the excluded disks, it follows that the entire function X(z) also tends uniformly to zero in the sal1}e angles, outside the excluded disks. Applying the maximum principle to Ix(z)1 in each of these disks, we infer that x(re iUj ) --+ 0 as r ->- 00. Applying the PhragmenLindelof theorem to this function in each angle 0; ~ 0 ~ OJ+l (j = 1,2, ... ,n and On+l = 01 + 217), we find that X(z) is bounded in the whole plane and consequently X(z) == const, but since x(re i6 ,) ->- 0 as r ->- 00, we have X(z) == O. Therefore cJ> (z) qr (z) 55 j(z) 6 (z), consequently h~(Ij)+ h

<. H(a)

and

h
< H(fJ).

We now show that <1>(z) is a function of minimal type of the proximate order p(r). We construct the auxiliary function VC
204

UNIQUENESS, INTERPOLATION AND COMPLETI:NESS

[CHAP. IV

Then we may suppose that

C< min

H(fJ) - h.(fJ) cos pO

( 4.55)

since e cos pO

< H(fJ)

- h.(fJ)

Then the numbers bn = (an)Vc(a n ) satisfy (4.42), and by Theorem II the series f)

«I> (an) V c (an) I' (an) (z - an)

(z) = I "r~. an I .;;":

2p

converges uniformly outside the excluded ell-disks and tends uniformly to zero outside these disks as Izl -~ 00. As above, we see that the function 'f () z =

,» (z) V c (z) - .') ( z ) I(z)

is holomorphic inside the angle IfJ I <; 7T /2p and at most of normal type of the proximate order p(r). In addition, it follows from (4.54) that for fJ = 7T /2p we have h.(fJ) + e cos pO < H(fJ) and consequently for sufficiently small 8 > 0 the function ",(z) tends uniformly to zero as z ~ 00 outside the angles 7T /2p - 8 < fJ <; 7T /2p and - 7T /2p <; fJ < - 7T /2p + 8. By the PhragmenLindelof theorem it follows from this that ",(z) is bounded in the entire angle IfJ I <; 7T /2p, and consequently h,¥(&)

-< 0

or in other words h(lJ)
Since e is any number satisfying (4.55), we must have, at the point 00 at which the right side of (4.55) attains its minimum. h
< O.

By rotating the plane, we can make any angle of opening 'TTl p coincide with the angle 101 <;; 'TT/2p, and accordingly every angle of opening 'TTl p contains a rayon which From this and the fundamental properties of the indicator (Chapter I, § 16) it follows easily that h.(O) <; 0 everywhere, i.e., that CI>(z) is at most of minimal type of the proximate order p(r).

Sec. 5]

APPLICATIONS OF LAGRANGE INTERPOLATION SERIES

205

It remains to establish the conclusions of the theorem when «1>(z) is an entire function of at most minimal type of the proximate order per). To do this we use a device of V. G. Iyer's [1]. For any positive integer m the function [«1>(z)]m satisfies all the hypotheses of Theorem 12 and consequently can be represented by the series [~(a ))'"

= fez) ~ /' (an)(; :N

[41 (z»)m

all) •

,,~l

Under the hypotheses of the theorem 1«1>(a..)1 <; M (n = 1,2, ... ) and hence, with the notation co

X (z) = we can assert that

~ If' (an) 1\

".-1

Z -

a,ai '

I 41 (.z) 1m -< If (z) I MIIIX (z),

or 1

141 (z)1

-< M If(z)X(z)lm.

Finally, since m is arbitrary, The conclusion of the theorem follows. We note that the hypothesis (4.53) on the indicator of «1>(z) is essential. In fact, the indicator of fez) is equal to H(O). Theorem 14 is a generalization of the following theorem of N. Levinson [1]. Let {A .. } and {,u .. } be two sequences of positive numbers having densities D1

and D,,, i.e.,

lim ~=D

''"co ,""II and inf(I..1HI-l..n)

If an entire function ±ip... and

then «1>(z)

> 0,

'"

inf(\Jon+1-\Jo,,)

> O.

«1>(z) of exponential type is bounded at the points ±lll' V 2 a

a. < 'It Dl.+ D""

== const. 15

PROOF. The set of all points of the form ±l", ±ip... evidently forms an R-set with index per) = 1. The canonical function fez) of this set can be represented in the form fez)

a.

= If' (z) Hz).

a.

11 This theorem in tum is a strengthening of an analogous theorem ofV. G.lyer (I]. which involves the stronger hypothesis < .. min (D1. D,,) instead of < ....; D1 + D!.

206

UNIQUENESS, INTERPOLATION AND COMPLETENESS

[CHAP.

IV

where

n ( z: ), 00

1 (z)

=

"~1

1_

A" The functions rr(z) and 'P(z) are of completely regular growth, hq(O) = 71'DA Isin 01 and "v,(O) = 17 D,. Icos 01, and the indicator diagrams of these functions are, respectively, the segme'lts x = 0, Iyl ~ 17DA and y = 0, Ixl ~ 17DIJ. When functions of completely regular growth are multiplied, their indicator diagrams are added, and consequently the indicator diagram of f(z) is the rectangle I,: Ixl " 17D IJ , Iyl ~ 17D A• It follows from the condition (]~ < 71'V D~ D~ that the indicator diagram of <J>(z) lies in the disk with center at the origin and diameter less than the diagonal of the rectangle I,. Therefore for OJ = ±arc tg D,.! DA we have

+

h4>(~j)

< H(~j).

Condition (4.53) is satisfied and consequently <J>(z) == const. The theorem is established. The case when all points of the R-set lie on the real axis and p = I is not covered by Theorem 14, since in this case H(O) = H(17) = O. This important case has been discussed in a number of papers. The following theorem of M. Cartwright may be considered fundamental for this problem. THEOREM 15.

If

diagram has width 2k

an entire function of exponential type whose indicator < 217 is bounded at the real integral points, i.e.,

c-'- n = 0,

1, 2, ... ),

then it is bounded on the whole real axis, and

I/(x)1
We may assume without loss of generality that h,(17/'2) The proof of the theorem is based on the formula

= k.

=

00

I(n+z)=

sln7tz ~ (_ 7tW ~ v~

1)"

!(n+v)slnw(v-z) (v-z)3'

(4.56)

-00

which holds for entire functions of exponential type that are bounded at the integral points provided that k < 71' (k = h,(71'/2) = h, - (17/2» and 0 < w < 71'- kY 11 It follows from this theorem that a function of degree zero that is bounded at the integen is a constant. In this formulation the theorem was first given by G. Polya. 17 This interpolation formula was introduced by Boas and Schaeffer [IJ and S. N. BernStein [lJ (see also Boas [ID.

Sec. 5]

207

APPLICATIONS OF LAGRANGE INTERPOLATION SERIES

To derive this formula we fix an integer n and consider the function of two variables r.p (C. z)

=

fen

+ C) -sin (~_ z) 01

(e -

Z)

01

00

_

sln7t~

~

7t0l

~

(-I)" !(n+'1)slnOl('1-z) z) (~- ... )

('I -

00

'1= -

When z is fixed this is an entire function of finite degree in ~, vanishing at all integral points. Hence after dividing this function by sin 1T~ we again obtain an entire function of exponential type in ~: t~(r T~'

z)- !(n+C) slnOl(C-z) -

sin 7tC

w (C -

~

'"' (_ If 00

+~ 7tw

~ ~=

!(n+'1)slnOl('1-z) ('1-C)('1-z)

-00

= ±1T/2.

The indicator of the first term is, for 0

(-+- ;)=k+oo-1t < o.

h

Since the indicator is continuous it is negative in some neighborhood of the points ±1T/2. The second term tends to zero on any ray arg ~ = 0 for 0 ~ 0, 1T. Hence 'P(~. z) tends to zero on any ray in a neighborhood of the rays arg ~ = ±1T/2. By the Phragmen-LindelOf theorem it follows that 'Pa, z) == O. Hence we obtain the identity

fen

+:) slnw(C-z) U)

=

(C - z)

00

sln7tC r.:w

~ (_

~ 'J=

-

If

!(n+'1)slnw('i-z) , ( ... - z) (C - ,,)

c..o

from which (4.56) follows by taking the limit as ~ -- z. Now put z = x ± i (-! ~ x ~ i) and pote that If(n 1,2, ... ); from (4.56) we easily obtain the estimate

+ v)1

~

M (±v = 0.

co

M

If (n + x -+- i) I < 7tw

ch r.

~

C,l 00

~ I 'i _

1 X

+ 112

-co

< -M7t ch 1t ch 00. w Since n is arbitrary, it follows from this estimate that for -

1/(x-+i)1 < P(oo)M ( P(oo)

=

1t

ch 1t

ch w

-",-,

00

< x < 00

208

UNIQUENESS, INTERPOLATION AND COMPLETENESS

[CHAP. IV

Phr~gmen-Lindelof

theorem for a strip (Chapter I, § 14) we find that C;;;; I and, in particular, on the real axis. The theorem is established. Our estimate for C(k), By the

f(z) satisfies the same estimate in the strip Iyl

C(k)-<: min P(III)=7:ch1t

... < .. -II:

Ch('It-k),

'It-k

is not exact. I8 Results analogous to Cartwright's theorem have been obtained not only for functions bounded on a sequence of points but also for functions that increase on a sequence. lS In particular, if the points of an R-set with p = 1 are sufficiently densely distributed on the positive real axis, the growth of the function on this sequence is the same as its growth on the positive real axis.THEOREM

(a}

16.

Let the sequence of points {an} on the positive real axis I), i.e.,

< a z < ...) hare density ~ with index p(r) (p = lim

..-... "" and for some d

n(r) - A r P(r) ,

>0 l-p (ani

,

an+l - an> dan and let F(z) be a holomorphic function for Re z In I F (z) I \-:1m P (r)

,. ... ""

< 00

~

0 such that

(Re z

> 0)

r

and

then

" Estimates f()r C(k) have been given in several papers. The most precise estimates are given in S. N. Bernstein's paper [2]. For the case k = 1T - lin, where n is an integer, Bernstein gives an exact estimate. As r1 ..... 1T Bernstein [2] gives the asymptotic estimate C(k) ~ 21fT In 1/(1T - k). A number of investigations have been devoted to various generalizations of Cartwright's theorem: R. Boas and Schaeffer [I), Duffin and Schaeffer [1]. S. N.Bernstein [2]. N. I. Ahiezer and B. Ja. Levin [I]. S. Agmon [1]. The paper of Plancherel and G. P61ya [1] belongs to the same circle of ideas . • 9 O. S. Agmon [1]. N. I. Ahiezcr [4] and B. Ja. Levin [16]. 10 The question of the connection between the upper limits

1-1m. In I F(r),

r ... ""

r,(r)

has been Jhe subject of a number of investi~tions. We mention the work of V. Bernstein [I]. A. Pfluger [3]. N. Levinson [1] and R. P. Boas [2]. The most complete results were obtained recently by W. H. Fuchs [1].

Sec. 5] PROOF.

209

APPLICATIONS OF LAGRANGE INTERPOLATION SERIES

The set of points ±a" (n

function

=

1,2, ... ) is an R-set with canonical

g (I - :;) co

f (z) = and indicator

H(fJ) = 1t.:1lsinfJ I.

(4.57)

Let - . In IF(a,,) I c= 11m ,,-+ 00

a P (an)

(4.58)

•

fa

Take a sequence of positive numbers iJt and construct the auxiliary function

< iJ2 < ...

•

with density at

> C'II'-l

00

IT (1 + z:)

V (z) =

n=1

'""n

with indicator hv (tJ) = 1t~11 cos 61.

(4.59)

From (4.58) and (4.59) we easily obtain that for some large n and consequently for some rl

E

> 0 and all sufficiently

> 0 and all n > n'l

In I F (all) V-I (a,,)!

< --- '1ja~

(lin) •

On the other hand, it follows from (4.57) and (4.43) that ' 1n ll'(an\ >l_0 I1m _ , n~oo

,1II (an

The last two relations imply the uniform convergence of the series

outside the excluded Cit-disks, and the sum tp(z) of the series tends uniformly to zero as Izl -+ 00 outside these disks. The function

Z (z) =

V

~:j(Z)

- y (z)

is holomorphic in the angle larg zl < 'TT/2 and at most of normal type of the proximate order per). We shall show that this function is bounded in the angle larg zl <; q < 'TT/2, To do this we first remark that without loss of generality we may suppose h~-'TT/2) = hrA,.'TT/2) and that by hypothesis h~'TT/2) < 'TTa.

210

UNIQUENESS, INTERPOLATION AND COMPLETENESS

[CHAP. IV

The indicator of the function F(z)

=

o:r (z)

V (z) f (z)

is equal to hF(O) - Tr~ Isin 01 - Tr~I Icos 01, and consequently for 0 close to

Tr/2 or -Tr/2 it takes negative values. Hence for sufficiently small () > 0 the function q (rei") tends to zero as r - 00 and 0 = ±(Tr/2 - tJ). Since 'f'(z) also tends to zero on these rays, the same is true of x(z). Then, applying the Phragmen-Lindelof theorem for angles to this function, we obtain

Iz (z) I -< M

( I arg z

1-< ; -

0) .

To estimate the indicator of F(z) we may now use the equation F (z) =

Ix (z)+ 'Hz») V(z)j(z).

We have shown that Ix(z)1 is bounded, and I'f'(z) I is bounded outside the excluded disks. It follows at once that hp(fJ) <. '!till sin (j I 1t~11 C05 (j I.

+

Putting

f)

= C, we obtain

In constructing V(z) we could have taken CTr- i . Hence we obtain hp(O)

~I

to be any number larger than

<-c.

The opposite inequality follows directly from the definition of the indicator.

6. Completeness of Systems of Functions. The Connection between Completeness and Uniqueness A sequence: !n(z)} of functions holomorphic in a region G will be called uniformly com'ergent in this region if it converges uniformly on every bounded closed set contained in G. A system {V n(z)} of holomorphic functions is called complete in the region G if for each function fez) holomorphic in G there is a sequence of linear combinations Pk

~ cn, k rpn (z)

(4.60)

n=l

with constant coefficients which converges to this function uniformly in G. In other words, the closed linear span of the system {IPn(z)} coincides with the set of all holomorphic functions. If this condition is not satisfied, the system

Sec. 6]

COMPLETENESS OF SYSTEMS OF FUNCTIONS

211

is calIed incomplete. By Runge's theorem 21 the system of powers I, Z, Z2, ••• is complete in every simply-connected region. We remark that if {9?n(z)} is complete in G, then the linear combinations (4.60) will approximate uniformly any function that is analytic in an arbitrary simply-connected bounded closed region lJ that belongs entirely to G. In fact, it follows from the completeness of the system {9?n(z)} in G that each power z'" can be approximated arbitrarily closely on lJ by linear combinations of the form (4.60), and consequently any holomorphic function can also be approximated by these combinations. We now establish the converse, i.e., that if every function holomorphic in an arbitrary bounded simply-connected closed region lJ belonging to the simplyconnected region G can be uniformly approximated by linear comhinations of the form (4.60) then the system {IPn(z)} is complete in G. In fact, letf(z) be any function holomorphic in G. Consider a sequence of bounded closed simply-connected regions

DI cD2 c ... cDn c

...

"" ~ Dn=O n=1

and positive numbers El > E2 > ... > En > ... , lim'Hoo en = O. By hypothesis there are linear combinations Pn(z) of the form (4.60~ ~atisfying the inequalities (zED n • n=l. 2 . . . . ),

If(z)-Pn(z)l<sn and consequently co

f(z) = PI (z)+ ~ [Pn+l (z)-PlI (z»), n=1

where the series converges uniformly in G. Let G be simply-connected and let G not contain the point at infinity. Let ~u denote the set of functions each of which is holomorphic in some simply-conilected region containing the complement of G and vanishes at infinity. Each element of 9Jl o is holomorphic in a region containing the boundary of G and infinity. Hence to each functionf(z) in 9Jlr; there is a rectifiable contour L t , belonging to G, such that!(z) i& holomorphic outside and on it. We say that a function f(z) belonging to IDlo is orthogonal to a function 91(z) holomorphic in G if (4.61) f(z)~(z)dz=O.

f

(Lfl

THEOREM 17. A system of functions {9?n(z)} is complete in G if and only if the orthogonality of a function f(z) of9Jlo to every function of the system {1P,,(z)} implies !(z) == o. 11

Cf. for example, A. I. Markulcvif [1).

212

UNIQUENESS, INTERPOLATION AND COMPLETENESS

[CHAP.

IV

PROOF. Suppose first that G is the unit disk. Let {qu(':)} be incomplete in this disk. Then there exist a disk Izl ~ r < 1 and a power z'" which cannot be approximated in this disk by linear combinations of the form (4.60). But then this power cannot be approximated in mean square by linear combinations (4.60) in any disk of the form Izl ~ R (r < R < 1). In fact, let

f

1

"I. :m - ~ C'lk~" (:) 12 d: < e'3. n=l

(CR)

By Cauchy's integral formula p.

zm -- ~

Cn.tfl,1

f [-

1

(z) = 21t/

n=1

r]

p.

CUR)

n=1

and by applying Schwarz's inequality we find that for p.

\ zm-

de

~ Cn.~" (,) C_ z'

,m -

Izl

~

r

I

~ c/I.C?" (z) < RR>_. n=1

We introduce the usual notion of scalar product for functions defined on the circumference Izl = R: 2n

(f. 'f) =

J

~- f(R.ei~) tfl (Re iB ) dfl o

= _1 2r./

r J(z)ip (R2') dz Z

w

(~

Z

= arg z).

(4.62)

(aR)

We orthogonalize the system {IPn(z)} and construct the Fourier series for zm co

Zm(z)

= n....;. ~ (z''', '~I!) Yn (z), t

(4.63)

where {1f!n(z)} is the orthogonal system equivalent to {lfin(Z)}. By the RieszFischer theorem this series converges in mean square on Izi = R to a function of integrable square. It follows from Cauchy's integral formula that (4.63) converges uniformly in each interior disk. Therefore the function Xm(z) is holomorphic in Izl < R and belongs to L2 on the boundary. The function co

is not identically zero and is orthogonal to all the functions {q: n(z)} in the sense (4.62). Putting

R2)

- I, - -1 = f (Z ) - F,z z

co

~

k-o

_

akR2/t zk+1 '

Sec. 6] we find that

COMPLETENESS OF SYSTEMS OF FUNCTIONS

J

/(z) '';-I! (z) dz

=

0

(n

=

I, 2, ... ).

213

( 4.64)

'e) p

Therefore we have constructed a function fez) E 9J1 u that is orthogonal to all the functions of the incomplete system 22 {IPn(z)}. We now turn to the general case of an arbitrary simply-connected region C, not containing the point at infinity and not the whole plane. Let z =O(w) be the function that maps the unit disk of the w-plane on the region C in the z-plane. It is clear that the completeness of the system {ern(z)} in C is equivalent to the completeness of {ern( O( w»} in the unit disk. Let the system {q n(z)} be incomplete. Then by what has been proved there is a functionf(w) which is holomorphic outside some disk Iwl > R (R < I), is not identically zero and satisfies

i few) ~,t (f' (w)

dw = 0.

(4.65)

WI o

Under the conformal mapping z = {few) the ring R < PI < 111'1 < pz < 1 is carried into a doubly-connected region r, which belongs to C and is bounded by two analytic contours Ll and L2 with equations Iw(z) I = PI and Iw(z)1 = P2' The region bounded by L2 belongs entirely to C, and the exterior region bounded by Ll contains the entire complement of C. The function f(w) which is holomorphic in the ring PI ~ Iwl ~ P2 is carried into the function F(z) = f[a'(z)], which is holomorphic in r, and equation ~4.65) can be written in the form

i F(z) '.pll (z) w' (z) dz

(4.66)

= 0,

(I'pl

where Lp is the image of the circumference Iwl

=

P (PI

< P < P2)'

The function

(I' (z)

=

r, can be -

,_I 'bol .

represented in the form

I' ~J)

IL,)

(:)

l, - -

z

d: + 2,,[ ~

rC- <0 d:.

~

'I'

Z

(L,

The,second integral, which we denote by 6(z), is holomorphic in the region bounded by L 2 • Hence 9 .• (z)') (z)dz = O. (4.67)

f

(L p '

22 It is clear that instead of the unit circle we could have taken a circle of arbitrary or even of infinite radius.

214

UNIQUENESS, INTl:RPOLATION AND C'OMPLETl'!IoIESS

[CHAP. IV

The function <J>(z) cannot be holomorphic throughout the interior of L 2 , since then /(II") would be holomorphic in the extended plane and zero at infinity, i.e., identically zero. Therefore V,(z) = <J>(z) - O(z) cannot be identically zero. If we take (4.67) into account, we can write (4.66) in the form

J~,,(z) '1

(z)dz = O.

(4.68)

Thus if {q n(z)} is not complete in G, there exists in 9J1/i a function tp(z) which is not identically zero and is orthogonal to every function of the system. Now let {9"n(z)} be complete in G and let tp(z) be an element of IDlu which is orthogonal to every function of this system. We show that tp(z) == O. In fact, since {Il'n(z)} is complete it follows from (4.68) that

Jzm'f (z) dz =

O.

(4.69)

CLpl

In (4.69) we can replace integration along Lp by integration along the circumference of any larger circle. Outside some circle we have

and we obtain from (4.69) that am = 0 (m = 0, 1,2, ... ) and consequently tp(z) == O. The theorem is established. 23 This theorem establishes the connection between completeness and uniqueness. In fact, by this theorem the completeness of the system {tp ..(z)} in G means that a function tp(z) E 9J1a is uniquely determined by the sequence of values of the functionals f'J"

=

J .~

(z) C? ' (z) dz,

(n = I, 2, ... ).

ILpl

Therefore every theorem on completeness generates a theorem on uniqueness and conversely.

7. Theorems on the Completeness of Some Systems of Entire Functions In this section we use the general theorem on the connection between completeness and uniqueness to obtain some theorems on the completeness of systems of functions in the corr,plex domain. U

For the case of a disk this theorem was proved by A. I.

Markulevi~

(2).

Sec. 7]

215

COMPLETENESS OF SYSTEMS OF ENTIRE FUNCTIONS

We shall say that the sequence offunctions {lPn(z)} is generated by the function F(z, A) and the sequence {An} if 'Pn(z) = cnF(z, A,,),

where Cn is a constant.

Here we suppose that F(z, A) is holomorphic in z in a region Gz if;' belongs to a region GA, and holomorphic in A in GA if z E Gz.24 In particular, GA may be the whole plane, and then F(z, A) is an entire function of A. In what follows we shall suppose that F(z, A) is an entire function of A and impose restrictions on the growth of this entire function. We call F(z, A) a closed kernel with respect to z if when Y'(z) ~ 9Jl G the equation

for A E GA implies Y'(z) == 0. 25 Consider the integral ql (A) =

f F (z,

A) 'Hz) dz,

(4.70)

(Lq.)

where Y'(z) is ~ fUIlction of 9Jl a and F(z, A) is a closed kernel. If the sequence {An} is such that for a function of the form (4.70) the equations IP(A n) = 0 (n = 1, 2, ... ) imply lPo.) == 0, then clearly the functions F(z, An) form a complete system26 in G. Conversely, if a function IP(A) of the form (4.70) is not identically zero and vanishes at the points {An} the system {lPn(z)} is not complete in G. In particular, if the sequence {An} has a limit point in GA, the system IPn(z) = F(z, An) is complete in Gz• For a kernel of the form 27 F(z, A) = f(z/..) we can state the following simple criterion for closure in the disk Izl

< R.

•• This assumption does not restrict the generality of the sequence {IPft(z)}. In fact, let

{9'..(z)} be any sequence of functions holomorphic in G, Dft closed regions in G, lim n --

15.. =

00

G, and M. = max zED. iIP.(z)i. Then the function

A _ sin "A

~

F(z, ) - - - k. M 1r

11=1

-1)·IP.(z) 1(' _

fin

J\

) n

is a holomorphic function of z in G for each A and an entire function of A for z E G, and the sequence A.. = n generates the given sequence of functions . •• A. I. MarkuSevic calls such a function a "complete function." •• Instead of the functionals F(z,)..) we may choose any system of functionals 9'.(z) = L,JF(z, A)l. For example, we may take IP..(z) = F1")(z, A), and if for functions represented in the form (4.70) the equations L,,[q:>(A)] = 0 (n = 1,2, ... ) imply IP(A) = 0, then {IPft(z)} is a complete system in G . ., The first problem on the completeness of systems of functions f(A.z) in a disk (f(z), an entire function) was set by A. O. Gel'fond [2], who obtained a theorem on the completeness of this system of functions by using the expansion of f().z) in a Newton interpolation series.

216

UNIQUENESS, INTERPOLATION AND COMPLETENESS

If fez) is holomorphic in the disk expansion are different from zero, then for this disk. In fact,

2~'

IAI

Izl

the function f(Az) is a closed k/!; nel in <Xl

(z) dz =

tal p

~ akbkAk

k=o

00

(9(Z) =

and all the coefficients of its (4.7~)

r /(zA) '1

[CHAP. IV

~J:l

k-o

' P < R)

and from the orthogonality of tp(z) with respect to the kernel it follows that hI; = 0 (k = 0, 1,2, ... ) and tp(z) == o. Conversely, if one of the coefficients a... = 0, then the kernelf(zA) is orthogonal to the function tp(z) = Z- ...-I. If fez) is an entire function, then clearly the nonvanishiag of all its coefficients is a necessary and sufficient condition for the c10su r~ of the kernel f(zA) in the disk Izl < R when A belongs to any region. In what follows we shall need an estimate for the growth of the function 4>(1..)=

f

f(zA)
(4.72)

1·I~r

under the hypotheses that fez) is an entire function of type (J of the proximate order per) and that tp(z) is bounded on the contour of integration. From the asymptotic inequality In I f(zA) 1<

(a + i) (r 1lit (,. 1).11

(s

> 0,

z = r),

1 1

which holds for large A, it follows that

Inlf(ZA)I«a+ ;) r'IAI'L(rI A!) and by the fundamental properties of the function L(r) In !f(zA) I

< (0 +a)r' I AIPlI).11 •

From this and (4.72) we obtain 1 4> (A}I

< 21trMexp {(a+e)r'l Alpll).ll} (M = sup 1l\I(z} I)· I-I-r

Finally, since

£

> 0 it follows from this that h. (6) ~ ar P•

(4.73)

Sec. 7]

COMPLETENESS Of" SYSTEMS OF ENTIRE FUNCTIONS

217

This estimate makes it possible to establish the following theorem on completeness very simply. THEOREM 18. If an entire function fez) represented by the development (4.71) satisfies the condition . (4.74) (n=O, 1,2,.... ) at! =1= 0

and has type at most (j with respect to the proximate order per), and {An} is a sequence of complex numbers, then the system offunctions Cfl n (z)

= /(AnZ)

is complete in the disk with center at zero and radius R determined by the equation (4.75) The proof is by contradiction. Suppose that the system offunctions < R. Then by Theorem 17 there is a function 1p{z), not identically zero, holomorphic in the region Izl ~ r (r < R), vanishing at infinity and orthogonal to all the functionsf(A"z), i.e., PROOF.

f(A"z) is not complete in the disk Izl

I /(Anz)·Hz)dz=O. Iz/=r

Therefore in this case the function «I>(A) defined by (4.72) vanishes at all points of the sequence {Aft}' It follows from Theorem 2 and the estimate (4.73) for the indicator of «I>(A) that if «I>(A) is not identically zero then

r

n

n ~00 I A.n IP (I

~

lon Il '<:::

P

epar .

For r < R this is impossible. Therefore we have shown that «I>(A) = 0 and since the kernel f(Az) is closed,. tp(z) = 0 also. Thus the assumption that the system {f(A"z)} is not complete in Izi < R leads to a contradiction. The theorem is established. If we repeat the discussion but use Theorem 3 instead of Theorem 2 to estimate the density of the set of zeros of «I>(z), we obtain the following theorem on completeness. THEOREM 19. Let fez) be an entire function of type (j of the proximate order p(r); let all the coefficients of its expansion (4.71) be different from zero, and let {A,,} be a sequence of complex numbers. Then the system of functions cp,,(z) == /fA"Z) is complete in the disk with center at zero and radius R determined by the equation

(4.76)

218

UNIQUENESS, INTERPOLA nON AND COMPLETENESS

[CHAP. IV

Theorem 18 was proved by A. I. Markusevic [2], and Theorem 19 is a sharpening of a theorem of A. O. Gel'fond [2]. We note that when R = 00 in (4.75) or (4.76) the system of functions (4.74) is complete in the whole plane. If the function fez) = ~ anz n is such that the limit

.!.

n r __

lim ~ (/L)l! Ian

I=

(4.77)

(epa)P

exists, where (p(n) is the solution of the equation

n = r P(rl, the estimate (4.76) is, in a certain sense, exact. Indeed, in this case we can choose the sequence {An} in such a way that for every entire fUlldiollf(z) satisfying (4.77) the system {f(Anz)} is not complete in any circle with center at zero and radius larger than the number R defined by (4.76). To show this we choose a regular set {An} such that the canonical function (z) of this set has a constant indicator (h(O) = 0' cI». Then by Theorem 3 of this chapter we have (4.78)

Let 00

Using the coefficients of this expansion, construct the function ex>

7(z)= ~~. "-' a zn+l o n

This series converges outside a disk whose radius r is determined by the equation n

,";-_ r= lim r Icnl

lim

cp (n)

Vlc,,1

=(a

=11-+00

n-+oo n _

p.

at

n _

VI ani

).!.

lim 'f ('1) lr I a1l I 7l-+o::>

If we use (4.78), we can write this equation in the form

rPF =I -

l'1m p~,. n -+00 I ion

i.e., r

=

n IP II

, An II

R. The function (z) can eyidently be represented in the form

J fez),) 'Hz) dz

(a

> 0).

(4.79)

/zl=R-r'

Since (An)

= 0 (n =

I, 2, ... ) the function 'P(z), which is not identically zero,

Sec. 7]

COMPLETENESS OF SYSTEMS OF ENTIRE FUNCTIONS

219

is orthogonal to all the functions f(zAn) (n = I, 2, ... ). Therefore this system of functions is not complete2M in the disk Izi ~ R + e (e > 0). The proofs of Theorems 18 and 19 depend on the uniqueness Theorems 2 and 3. Completeness theorems can also be obtained from the uniqueness theorems discussed in § 2. We shall prove the following completeness theorem which depends on Theorem.7. THEOREM 20. Let Un} be a sequence of points of the right hand half-plane, lI(r) the number of points of this sequence in the disk Iz - r/21 < r/2, and

d --

I'l m v(r) -. -- r

(4.80)

"0+00

Then the system {e-~"'} is complete in every strip I of the form

Yo < 1m z

f e).z,~ (z) dz

(4.81)

(L)

is an entire function of exponential type and the width of its indicator diagram is less than 27Td. If V'(z) is orthogonal to all the functions of the system {e A".} then (An) = 0 (n = 1, 2, ... ) and, by Theorem 7 of § 2, (A) == O. The kernel ,;z is closed, and consequently the system {e An'} is complete inside any contour lying in I. The theorem is established. Putting w = e', we obtain the following theorem which contains a theorem of T. Carleman [1]. The system {zAn} is complete in the angle To < arg z < To + 27Td, where CPo is any number and d is defined by (4.80). The formula (4.81) can be inverted (Chapter I, §20). This makes it possible to establish the following sharp theorem. THEOREM

21.

If {An} is a sequence of complex numbers for which the limit d= lim ~ "0+00

An

(d

> 0)

(4.82)

exists, the system {e A".} is complete in any region G which every straight line parallel to the imaginary axis cuts in a segment of length less than 27Td, and the I I One can show that Theorem 18 is sharp in the same sense. To do this one must construct a function Cll(A) in the same way as the example in § 1 which established that the estimate in 1beorem 2 is sharp.

220

UNIQUENESS, INTERPOLATION AND COMPLETENESS

[CHAP. IV

system is not complete in any region which contains a segment of length 21Td parallel to the imaginary axis. 29 PROOF. We first establish the second part of the theorem. Let G contain a line segment I of length 21Td parallel to the imaginary axis. The inilicator diagram of the function

(4.83)

is a segment of length 21Td of the imaginary axis. For suitable choice of the number y the conjugate diagram of the function O().) = e;·A (A) coincides with I. All the singular points of the Borel transform V'(z) of O(A) lie on this segment. Let L be a contour surrounding the segment J and belonging to G. Then

o(i.) = 27tl -1 j~ eA.,!"' (z) dz ' (Ll

and the function y;(z) is not identically zero, is holomorphic outside and on L, and is orthogonal to all functions of the system {e A.-}. Therefore this system is not complete inside this contour, and therefore not in G. Now let G be intersected by every straight line parallel to the imaginary axis in a segment of length less than 21Td. If the system {eA.,} is not complete in G, there is a function V'(z) which is holomorphic outside and on some contour L that belongs to G and is orthogonal to all functions of the system. On the other hand, the entire function of exponential type 9 (i..)

= 2~1

reA.,;. (z) dz

(i)

vanishes at all points An. Hence it follows that the quotient •

Z (I.) =

Ii' (A) ,I> (i.)-

is a holomorphic function of exponential type in the right hand half-plane (Corollary 2 to Theorem 5 of Chapter III, §4). The function ~) is of completely regular growth. Hence under multiplication by it the indicators add and h, (6) = hq" (6)+ h1. (6) = h1. (6)+ 'ltd t sin /)

I

Therefore the indicator diagram of q;(,l) is obtained by translating the indicator diagram of X(A) by the vectors i1Td and -i1Td and taking the smallest convex region containing the two resulting regions. U While this book was being prepared for publication an analogous result was obtained by A. Leont'ev [4).

Sec. 7]

COMPLETENESS OF SYSTEMS OF ENTIRE FUNCTIONS

Thus we find that

,

h'f (+ 0) -

221

,

h~ (- 0) ~ 2Trd,

and consequently the boundary of the indicator diagram of q;(z) contains a segment I perpendicular to the real axis and of length not less than 2TTd. The ends of the .segment symmetric to I with respect ·to the real axis are extreme points of the conjugate diagram and hence are singular points of VJ
+

is a maximal region of completeness for the system {eA.'}.

CHAPTER

V

FUNCTIONS OF CLASS A

In applications of the theory of entire functions, an important role is played by those entire functions that satisfy the condition

~llm ~J
(5.00)

k,-l

where the ak are the zeros of the function. These functions are natural generalizations of a function with real zeros only. Indeed, it follows from (5.00) that the zeros {a~} lying outside the angles larg zl < band larg Z - 7T1 < () form a convergent series

~ I ~~I <

00

and hence that "almost all"' zeros {ak} lie in the "neighborhood" of the real axis. We shall refer to these functions as functions of class A. In an analogous manner one can define functions of class A in a half-plane (1m z ~ 0 or 1m z ~ 0) by assuming that the sum (5.00) is formed from the zeros of the function that lie in this half-plane. 1 The functions of greatest interest are the functions of class A of exponential type. From Lindelof's criterion (Chapter I, § II) 1:E lanl
r

~ o

.., 1 In

!

(I -

lnl F(x)F(-x)ld l+x3 x.

this case the condition (5.00) plays the same role as the condition of Blaschke

la.1>

plays for functions defined on the unit circle. One of these conditions can be

&=1

obtained from the other by mapping the unit circle on the half-plane.

222

Sec. I]

223

FUNCTIONS OF CARLEMAN

In § 1 we show that the bounded ness of this integral is equivalent to requiring that the entire function F(x) of exponential type belong to the class A.2 If F(z) is also of exponential type in the half-plane 1m z ~ 0, hF(O) + hF(-TT) = 0, and if the limit of this integral exists when R --+ 00, then F(z) is a function of completely regular growth. 3 For functions of exponential type, these two conditions are necessary and sufficient.' Besides the criterion mentioned in § 4, there exist other criteria which guarantee that a function of exponential type in a half-plane be a function of class A and be of completely regular growth. In particular, this will be the case if the integral

r

00

oJ

lolF(xlJ d 1

I

X2

X

-00

or 00

(x) I dx

F S10+1I+x

2

-00

converges. From the last criterion (which is due to Cartwright [1]) it follows that functions of exponential type, bounded on the axis (or belonging to U'( - 00, 00) (p > 0), are functions of completely regular growth. We shaH show that for functions of exponential type the convergence of the second of these integrals implies that hF(O) + hF(7T) = 0. 5 In this book we shall also investigate the properties of the indicator diagram of functions of exponential type and of class A, and we shall consider also certain other problems related to this class of functions. In conclusion, in § 6 we shall prove a theorem of M. G. Krein that is related to these problems: An entire function fez) of class A that admits the expansion 1

f(z)

~

=R(z)+ ~Ck k~1

ZP) , --=--+-+-2 + ... +p-.:t a

(liZ Z ak ak ak

k

where R(z) is a polynomial, is a function of exponential type and of completely regular growth.

1. Formula of Carleman. Criterion for an Entire Function of Exponential Type to be of Class A The next theorem is of prime importance for what follows. 8 • N. I. Ahiezer [2] and N. N. Meiman [31. a B. Ja. Levin [3].

• A. Pfluger [5]. • While the present book was in preparation, there appeared a book by R. P. Boas [I] giving a somewhat different presentation of these problems based on a formula of R. Nevanlinna [2). 6 T. Carleman [1].

224

[CHAP. V

FUNCTIONS OF CLASS A

THEOREM

If F(z) is a holomorphic function in the region

1.

o < A <: I z i < R

u 1m z>- 0

(5.01)

= rke'o. (k = 1,2, ... ,n) are its zeros in this region, thell

and if ak

To

~

(_1_ _ ~) sin 'J k = _1-

~

<,./c
r"

roN.

/(J

J' In I F (Re ta ) I sin f) d'J

u

A

+-2~-J (;2 - ~"i) In I F (x) F (--- x) I dx+ AdF,

R),

(5.02)

A

where (5.03) This formula, together with Jensen's formula and ~he formulas (3.06) and (4.20), establishes a connection between the distribution of the zeros of the function F(z) and the growth of the quantity In IF(z)l. PROOF. Let us first assume that the function F(z) has no zeros on the boundary of the region (5.01), and let us consider the integral

J=

2!1 f (~- :~) In F (z) dz, (a,

taken along the boundary of the region (5.01) in the positive direction beginning at the point A.:

_ i.

(5.04)

On the other hand, integration by parts yields 1 ,h. [( Z 1) J-= 27tl 'f a R2+Z

In F (z) ] -27t11 f F' (z) ( R2+z1 ) dz. J-"(z)

Z

(5.05)

(0,

Here, denotes the increase of the function during one complete passage over the contour C. This increase is equal to (A./R2 + l/l)n(R,l), where

Sec. 1]

225

FUNCTIONS OF CARLEMAN

nCR, A) is the number of zeros of the function F(z) within the region (5.01). The second summand in (5.05) is equal to

Equating the imaginary parts of (5.04) and (5.05), we obtain (5~02). If as R varies a zero occurs on the semicircle Izl = R, then the formula (5.02) remains valid because both parts of the formula are continuous functions of R. It is also easy to see that formula (5.02) remains true if there are zeros of F(z) on the axis of reals. REMARK. The quantity A,,(F, R) is bounded when R we have the limit

1

AdF, (0) = 1m 2-

f "

> A, and when R -+ ex:>

-iD

In F (Ae iO ) ~ d1j II

1t •

o

•

If the function F(z) is holomorphic at the origin, and if F(O) lim A). (F, R) = ).~o

= I, then

i 1m F' (0),

and we obtain the equation

~ (~- ~) sin Ok = r.~ "'. '" R

+2~

"

f

In IF (ReiD) I sin fJ dO

0

R.

J(;2- ~2)lnIF(X)F(-x)ldX+ ~

ImF'(O).

(5.02')

o

THEOREM 2. In order that an entire function of exponential type may be of class A it is necessary and sufficient that for some fixed A > 0 and for every R > A the following inequality be valid:

R

f !."

If(x)/(-x)1 dx x3

< M t.).,

(5.06)

).

where Mf,A is a constant.

It follows from this theorem, in particular, that every entire function of exponential type that is bounded on the axis of reaIs is of class A.

226

[CHAP. V

FUNCTIONS OF CLASS A

PROOF. Let us first prove the sufficiency. Suppose that condition (5.06) is satisfied. Setting x

y(x)=

flnlf(E);(-~)lda

.

~

).

and integrating by parts, we find that R

f (1- ~22)

In

If(X~(-- X)I dX=~2

A

R

J

4(x)x dx <Mp,

(5.07)

A

and hence that the second summand on the right hand side of Carleman's formula (5.02) is bounded. Furthermore, since fez) is a function of exponential type, it follows that for R > A and for some constant k"A we have In If(Rei~1 ..;;; k"AR, and 1t

r.~

J

In If (Re iO ) I sin fJ dfl

< ~ kr,

A'

(5.08)

o

Therefore, the function of R in the right hand part of the equation (5.02) is bounded. On the other hand, we have

(5.09) This last inequality follows from (5.02), (5.07), and (5.08). From (5.09) it follows at once thatf(z) is a function of class A. For the proof of the necessity of the condition, we make use of an estimate of the modulus of an entire function from below (Chapter I, § 8, Theorem II). From that estimate it follows that to every TJ > 0 there corresponds a number HlrJ) > 0 such that In I f(Re iU ) I > - HI ('fj) R (0 <. 0 < 2or) (5.10) for all values R > R" except possibly for values of R that lie in intervals the sum of whose lengths does not exceed TJR on any interval (0, R). Let the number R lie outside the exceptional intervals. Then tR ~

fo" In If(ReiD) I •

sin 0 d'J

> - ~HJ. ('fj). K

(5.11)

Sec. 1]

227

FUNCTIONS OF CARLEMAN

~A"'k"R sin

Iff(z) is of class A, then

()/r/c

< N, and

from (5.02) we obtain

J?

r Inl!(x~(-X)ldx
J?

r

+ ~3

In 1/(x)/(-x)ldx.

(5.12)

).

The last term in (5.12) is bounded because, for

x> A,

In IJ(x)f(- x) 1< 2kpx

(5.13)

and hence R

R;.f In If (x) / .( - x) Idx < k,. 1

I.·

(5.14)

o

Thus, if the point R lies outside the exceptional intervals, then R

r Inl!(x~(-X)ldX<M~.).,

).

where the quantity M".A does not depend on R. If R lies in an exceptional interval, one can find a point R' lying outside the exceptional intervals such that R' < Rand R - R' < 1JR. From the inequalities (5.13) and (5.14) we obtain

.r In I!(X~{(-.,t) dx =

f

).

I.

R

,

n

+

J?'

In I!(X~(-X) , dx

Inl!(x)!(-x)'dx<."M' r Ii,

f.

x2

I.

+2kf. l In(I+) "'l .

(5.15)

REMARK. In the proof of the sufficiency of the condition (5.06) we did not make use of the fact that fez) was an entire function, but we used only the conditions thatf(z) was holomorphic and of exponential type in the half plane 1m z ;> O. We thus established the following assertion: If a holomorphic function F(z) of exponential type satisfies condition (5.06) in the half-plane 1m z ;> 0, then F(z) is of class A in this half-plane. THEOREM

3.

For an entire function fez) of exponential type, the condition

n

J' >.

In I/(x)!(-x) Id XZ

x

< M(~) (.).

(a)

228

[CHAP. V

FUNCTIONS OF CLASS A

is equirolent to each of the following conditions:

I<

II

If

llll!(X)!(--X)1 ---'-x""'"a-'-dx

11)

M t.

(b)

l,

l

RI

.r

Inl/(x)f(-x)Idx<M(21l 1+ x 2 , ••

(c)

o R

Ior •

(the

In 1/(x)f(-x)1 dxl

1

+ Xli

< M(3\

(d)

,.

MJil, i = 0, I, 2, 3, do not depend on R).

PROOF. First, we shall prove that for every entire function of exponential type we have the inequality

J,

R In A

I/(x)f(- x) I d ~

X4

X

> M(.)t.'.

,

For this we note that the formula of Carleman implies that R

f

~

~

Inlf(x)!(-X)ldx> _ _ 1 flnlf(ReiO)\sinOdfJ-A.(f,R) X2 1tR • 0

+ ~a ~Rln \f(x)f(-x) Idx. Jfwe now make use of the bounded ness of AJ.(f, R) and of the inequality (5.08), we obtain R

f

In

I!(X~(-X): dx > Mj~~ + ~2

~

J If R

In

(x) f(- x) Idx.

(5.16)

A

Next, setting

J~ (IJ) =

f

R

In I /~xei8ll dx

l.

and integrating by parts, we find that R

~

f

Inlf(x)f(-x)ldx

l.

= ;1Ji"'(O)+Jf(1t»)--~

R

f IJf(O)+Jf(1t»)dx. ).

(5.17)

Sec. I]

229

FUNCTIONS OF CARLEMAN

In Chapter III (Lemma 4) we established an inequality which for entire functions of exponential type has the form

I Jf (~) I < cr. AI x I

(cf.A is a constant).

(5.18)

From this inequality and (5.17) we obtain R

I~2 J

In If(x) f(-x)1 dx

1< 3cp,

).

and after substitution into (5.16) we have R

f l

> M.l + 3c~A =

In 1/(x)/(-x)1 dx ~

M(~I

~l.

Combining this with (a) we obtain (b). Thus, the conditions (a) and (b) are seen to be equivalent. For the establishment of the equivalence of the conditions (b) and (a) with those of (c) and (d), we consider the integral R

J(~-I~X2)lnlf(x)f(-x)ldX - f +)

l

R

-

x(1

X2)

In I/(x)/(-x) I d x X.

l

Integration by parts yields R

f (! -

. l

x~

_1_.•) In If(x)f(-- x)1 dx 1 +x~ R

=

R(I

J

~ R~) (J~ (0) + J~ (TC») + X23(~~:~2)~ (Jr (O)+Jf (TC») dx. l

Next, applying the inequality (5.18) we obtain, for R

> 1,

R

If G2-1~X2)lnlf(X)f(-x)ldxl<4c,.l' l

Rewriting this inequality in the form R

f

Inl/(x)/(-X)ld

x2

X

-4

< C,.l

l

f

R Jnl/(x)/(-x)1 d

1 +x2

X

l

R

<

f

l

Inl/(x)/(-x)1 Xl

d +4 x C,.l,

230

FUNCTIONS OF CLASS A

[CHAP. V

we easily deduce the equivalence of condition (a) to (c), and of (b) to (d). This establishes the truth of the theorem. Thus each of the conditions (a), (b), (c) and (d) is necessary and sufficient that an entire function of exponential type, belong to class A. The integrals occurring in these conditions play an important role in the criteria that an entire function of exponential type not only belong to class A but also be of completely regular growth. For the establishment of these criteria we shall derive a special representation of an entire function of class A. This representation is analogous to a known expression for a holomorphic function bounded in a circle. The next two sections are devoted to this representation.

2. Representation of a Function Harmonic in a Half-Plane In this section we establish some sufficient conditions for the representability of a function harmonic in a half-plane as a Poisson integral

1 (N. G. CEBOTAREV [6». An entire rea/7 function fez), satisfying the condition 1m fez) > 0 when 1m ~ > 0, has the form fez) = a.z + fJ (IX > 0, fJ is a real number). LEMMA

PROOF. From the principle of symmetry, and from the hypotheses of the lemma, it follows that Imf(z) 0 when 1m z O. The increase of argf(z) on any circumference does not exceed 21T. Hencef(z) has at most one zero. This single zero Xo is real. The function q:>(z) = (z - xo)-lJ(z) has no zero, and larg q:>(z)1 .;; 1T in the whole plane. Hence the entire function w = In lP(z) maps the entire plane z on the strip 11m wi <; 1T, and therefore q:{z) = const. 8

<

LEMMA 2. the condition

<

If a continuous real function u(t), defined on the real axis, satisfies -.,u (-:t-t)= 0 1m -

t

t-.:tco

< 00

(5.19)

dt

(5.20)

and if the integral co

J= ! 1t

J' 1u+(/)

-co

t3

• An entire function is said to be real if it takes on only real values on the axis of reals.

• It is sufficient to note that the function" ... __1_ 2 -, is bounded on the entire plane z w - ,,' <1,,1 < ,,) and to make use of a familiar theorem of Liouville.

Sec. 2]

231

FUNCTION HARMONIC IN A HALF-PLANE

exists, then the integral ro

Y

v (z) = 1t

f u (t) (I _

dt

+ y3

X)2

(5.21)

-00

will exist for all real values of the variables x, y (y

~

0).

We shaH first prove that conditions (5.19) and (5.20) imply the existence of the integral PROOF.

=

JI

u ( -+- t) I~ .

(5.22)

1

Setting x

J

9(x)=

U(t)'';f2'

o

we obtain from the existence of the integral (5.20) the inequality 1V'(x) I < x < (0), and for every a> 0 and h > 0 we have

~

M

( - 00

u+n

f U ~ = r'HI) (~+ f) r+ (I)

a

a

a+h

h+ r '1 (I) (k + I~) dl = J

0

(± ).

We have thus established the existence of the integral <Xl

f U(-+-t)t- 3 dt. 1

On the other hand, it follows from (5.19) that9

f=

u+(+-t)/-Sdl

<

D::l.

1

This implies the existence of the integrals t) dt = j.u_ (:±: II

=

I' u ~:±:

t) -- u +

•

fS

(:±: t) dt '

1

J.

and, since lu(t)1

_

= u+(t) + U_(I),

also the existence of the integral (5.22).

• The functions u+(t) and u_(t) are defined as usual by u+(t)

> 0, o when u(t) '" 0, -u(t) when u(t) < 0, { o when u(t) ;> O.

={

u_(t) =

U(t) when u(t)

232

FUNCTIONS OF CLASS A

[CHAP. V

Furthermore, the equation Y

(I - x),

+ y2 -

f2

Y 0+ 1 = (t 9)

and the existence of the integral (5.22) imply that the existence of the integral (5.20) is equivalent to the existence of (5.21) for all x and y. This completes the proof of the lemma. In the next lemma we give an estimate of a function represented by Poisson's integral. 3.

LEMMA

The integral (5.21)

defines a holomorphic function L'(Z) in the half-plane y > 0 whose limiting values on the real axis coincide with the L'alues of the function u(x), and which satisfies the inequality I,v(z)

.

where on E.

E

I< C

• _L .-

E~ --

:Y I

(r =

I zl).

(5.23)

is an arbitrarily small positiL'e number, and c. is a constant depending The existence of the integral (5.22) implies that the integral

PROOF.

(.0

f'

[

1 ~ u (I) (I _ xp y v (z) -- ly = 1t

+ y~

1

-10"

~ 13 ] dt

(5.24)

converges uniformly on any closed subset of the half-plane y > 0, and therefore represents a harmonic function for y > O. The equation v(x) = u(x) is established in the usual way. Let us now proceed with the proof of the validity of the inequality (5.23). Let us set " (I) when when

Z..,(t) = { 0

and

It I > N, It 1<: N,

(N) 0)

:J:

'~N (x) :=

.f ZN (t) t-

2 dt.

(5.25)

o

Since 1/11

I"NC/)I

-

< '1.

I/tl

+I=

O(r"), it is true for every '1

>0

and N> N, that

Sec. 2)

233

FUNCTION HARMONIC IN A HALF-PLANE

Integrating (5.21) by parts, we obtain 'V

(z)

CoO

+"ityf''IN (t) iiid[

t + y~ x)2 3

(t _

]

(5.26)

dt.

-00

When r

> 2N, the following inequalities hold: N3

x)~ + y~

(N -

N3

< I,

(N+x)lI+ya< 1.

Therefore the absolute value of the expression in braces of (5.26) is less than 417.

d[

For the evaluation of the third term of(5.26) we note that dl (t _ changes its sign when t = 0 and t =

12

X)2

+r ]

x + y/x, and hence we have the result

00

00

f tjlN(t)ft[(t_~;2+y~]dt -~1j JI~[(t_:;a+y~]ldt

-00

-00

With the aid of these inequalities we derive from (5.26) the inequality

I 'V (z) I < 0

(1)

+ ~ ~Y. It

This completes the proof of the lemma. THEOREM 4. In order that a function u(z) which is holomorphic in the halfplane y > 0 and is continuous in the closed half-plane y ;> 0 and also satisfies the condition

u(z) \-:1m - -

(a)

'_'-+00

I zl

< 00

be representable in the form

J' CD

(b)

u (z) =

~

-QD

dt

u (t)(t-x)J

+ y,+ky

(z=x+ly).

234

[CHAP. V

FUNCTIONS OF CLASS A

where

lim u (iy) >

k =

(c)

Y

/(-++00

-00,

it is necessary and 3ufficient that the integral

Jf + U

(I)

dt

t2

be convergent. 9a

PROOF. The necessity is obtained directly from the equation (c) if one sets z = i. For the proof of the sufficiency we note that from Lemmas 2 and 3, and from the representation of the function (5.21), it follows easily that the function (5.27) w(z)=u(z)-v(z) - k+ y (k+ = max(k, 0» is harmonic wilen y

> 0 and satisfies the conditions

~i)

w(x)=O.

(2)

W

(3)

'W

(iy) <;;

0

(Z) <;;

0 (r)

(y),

+ -y-. a (rZ)

From condition 0) it follows on the basis of the principle of symmetry that the function w(z) can be extended harmonically into the lower half-plane as an ;Jdd function of y. The harmonic fur:ction y+h ,+h Wh

(z) -=

4~2

Jf

w (x+ it) dt d't

y-h .. -h

~atisfies the conditions (1) and (2), while condition (3) is replaced by the condition

(3' )

By the application of a theorem of Phragmen and LmdeloPo inside the angles 0"; arg z ..; 'fT/2 and 'fT/2"; arg z ..; 'fT, we find that Wl\(z) - ey is bounded in the half-plane y ;;. 0 and that Wl\(x) = O. Applying agam the theorem of Phragmen and Lindelof to the function wl\(z) - ey in the half-plane y > 0, we obtain (y:;> 0). '-This representation of a harmonic function in a balf-plane was established by R. NevanIinDa [1] under the hypothesis that

tit f _"""" I"+(t) + 12 <

00.

11 The theorem of Phragmen and Lindelof for harmonic functions can be obtained from Theorem 22, § 14, Chapter I, by noting that a harmonic function is the logarithm of the modulus of a holomorphic function.

Sec. 2] Since

£

FUNCTION HARMONIC IN A HALF-PLANE

>

235

°

is arbitrary, Wh

(z) -:; 0 when 1m z? O.

Next, taking the limit as h - 0, we obtain the inequality

°

>

w(z)< when 1m z 0. Denoting by X(z) the conjugate harmonic function, we construct the entire real function If> (z) = X{z) -lw (z),

which maps the upper half-plane into the upper half-plane. By a lemma of N. G. Cebotarev this function has the form tp(z) and hence w(z}=-ay.

== a.z + p

Substituting this into (5.27), we obtain u(z)

=

+ (k+ -a)y. k+ - a > - 00. This proves the theorem.

v(z)

It follows from Lemma 3 that k =

lim w (ly) = O.

Y

II-++co

== 0, and w(z) == O. This proves the theorem. COROLLARY 1. If the function u(z) is harmonic in the half-plane 1m z > 0,

Therefore ex

is continuous when 1m z

(a)

> 0, and satisfies the condition ',.u(z) < 00 1m - IZ I

,.,-+00

and if the integral

co

f

-co

U

(I) dt

1+12

converges, then '0

(re~)

< <

= k sin 0 (0 0 'It). (5.28) r This corollary is obtained directly from the representation (b) and from Lemma 3. lim

U

,.-+00

COROLLARY

when 1m z (c')

2.

If the function u(z) is ha.rmonic when 1m z > 0, is continuous

> 0, satisfies the condition (a) and the conditions co (I) + u ( - I) dt .I 1 +t3 exlS

f

II

$

o

and (d)

(x) ~ 0 I x I "'" •

lim " Izl-+oo

236

FUNCTIONS OF CLASS A

then

'8

lim u (ret) ,. ... 00 r

= k sin 0

(0

PROOF. By Theorem 4, the function u(z) form (b), and therefore ;8

[CHAP. V

< /j < 'It).

(5.28')

+ u( -z) is representable in the

+ u (rei(~-8,> = 2k sin fJ

< <

(5.29) (0 IJ 'It). r Furthermore, condition (d) implies that the indicator diagram corresponding to the function u(z) is an interval (segment) on the axis Oy, namely, lim u (re)

,.-+00

-I' ' I'J. 1m u (re"") = ksm

(5.30)

r

,.-+00

On the other hand, the relations (5.29) and (5.30) imply that lim u (re,q) = lim u (rei'l) + u Cre '\It-~» lim u (re I(It-'") ,.~

r

r

,.0+00

r -+00

r

=ksin/J. This result and (5.30) imply equation (5.28').

REMARK. The results of this section clearly remain true if we replace the hypothesis that u(z) is continuous in the upper half-plane by the hypothesis that u(z) is harmonic in y ;> 0 except perhaps for isolated logarithmic singular points on the real axis. 3. Representation of a Function of Exponential Type and of Class A in the Upper Half-plane If a function F(z) of exponential type in the half-plane y ;;> 0 has no zeros in this half-plane, and if the integral

f

In I F (t)

-00

t2

+1

I dt

converges, then Theorem 4 of the preceding section implies. that this function has the representation

-00

where k == h~:"/2). In order to derive criteria for the complete regularity of the growth of functions of class A and of exponential type, we shall need a representation of such functions in the more general case, when the function has zeros in the upper half-plane. In this section we shall obtain such a representation. First we shall prove two lemmas. LEMMA

4.

If Ihe series (5.31)

Sec. 3]

237

FUNCTION OF EXPONENTIAL TYPE

converges, then (he infinite product co

x(Z) =

II(l-~)(l-_Z k=1

ak

)-1

(5.32)

ak

converges uniformly and absolutely in any finite Closed part of the plane that does not contain points ale' PROOF.

Let us write X(z) in the form X(z)=

n[l +Z(l-!' )-1(_1 __1 )]. k=1

ak

ak

(5.33)

a"

By a well-known criterion, uniform and absolute convergence of the infinite product (5.32) is equivalent to uniform and absolute convergence of the series

~) (1 - a: r 111m :"', Z

1

while the convergence of this series is equivalent to the convergence of the series 1::'..1 11m 1/01:1. This establishes the lemma. LEMMA 5. If {ale} is a sequence of complex numbers such that 1m air > 0 and 1:f 11m 1/01:1 < 00, then lim. In Ix (re4t) r = 0, r-+cO

r

where X(z) is a meromorphic Junction defined by the equation (5.32}.11 PROOF. We note first of all that the modulus of each factor in the product (5.32) is less than unity when z lies in the upper half-plane. Therefore,

I X(z) 1< 1

(1m Z > 0),

lim In I X(re4t) I <: 0 r

(0

and hence r-+oo 11

< 6 < 1t).

(5.34)

B. Ja. Levin [5], N. N. Ahiezer [2]. In the article [1] of Ahlfon and Heins the following

. lact ~.IS proved: t he I'Imp.... .., In more precIse

rX<,,">1 = 0 under·..,e cond'Itlon . t hat r goes to L

r

infinity without taking on values in some set E of finite logarithmic length, i.e., a set E such that

i.x-< dX

00.

We <:
measure, but not every set of zero relative measure bas finite logarithmic length. Hayman [1] eatablished the stronger result that (ln1x(re")I)/ r ::: 0 as r ..... 00, r E E, 0 < II < fl. Azarin [3] noted that the exceptional set E can be replaced by an exceptional set of disks ~ such that IJ~/ 'j < 00, where ~ are the diameters of the disks and 'J are their distances from the oriain. We shall use Lemma S in what follows, but replacing it by the theorems of Hayman and Azarin 1eads to somewhat stronger results. In the cited paper Azarin extended the result to functions of n variables that are subhannonic in a cone. [Tran.rlator's remtlTk: See also J. L. Doob, [1].]

238

FUNCTIONS OF CLASS A

[CHAP. V

Next, we need to find a lower bound of Ix(z)l. For this purpose we express X(z) in the form

x(z) =

Xl (z) X2 (z),

(5.35)

where

Let us first examine the function Xl(Z), For this purpose we represent Xl 1(z) in the form

+ xl "

From the inequality In 11 -

111

When larg z -

Izdz) I -< 2"

Ixl it follows that

2"-~<

7T

/21 "

7T

I

~"" 11 ~Zll

arg "k-2" \ < 2"

1m

ak

I I -k I Z I " (sin 8) -\ k > 0 we have

/2 - 28, the inequality

and therefore within this angle and for some N

1 ;k I·

Q

Q

holds,

For a given E > 0, and for a sufficiently large N, the right hand side will not exceed E Izl/4. After N has been chosen, one can select the number R. so that if Izi > R. then the following inequality holds:

" I

~

IZ~

"I"
l,m;kl<:lzl.

z-~< argak-i

We have thus proved that if Izl

> R"

11'/2 - 2d

In I Xl (z)j

-

then within the angle larg z - 11'/21

<

II

I z I,

<

Sec.

31

239

FUNCTION OF EXPONENTIAL TYPE

and therefore In I II (rei~) I . . . . 0 . \ 1m -:? •

r -+ ex>

(5.36)

r

Let us now consider the second factor in (5.35). All the zeros of the function - 11/21 ~ 11/2 - J. Therefore they satisfy the inequality 1m at ~ sin ~ lOti, or X2(Z) lie within the angle larg Z

I lm-I-I~~· ak I ak I ,;?'

From this, and from (5.31) we see that the series

~

Iarga,,-i/ < -i- 3

TtikT

(5.37)

converges and hence that tp(z) =

IT (1-~) Iargak-Z-"I ..:: i-~ all

is an entire function of genus zero. Therefore Theorem 2, Chapter II,

r1m

In I ~ (rei")

I

= 0,

~9'

= 0,

and by

=0.

r

r-+ex>

~cp(O)

where r tends to infinity without taking on any values of a set E of zero relative measure. The function

is also of genus zero and has no zeros in the upper half-plane, and hence lim In 1'4' (re'~) 1_

r

when 0

~

=0

0 < 11. Since ., (z) = 11.9

!.(z)

'4' (z) ,

the last two equations imply that lim In 11.2 (re'8) r-+co rEB

r

I = O.

240

FUNCTIONS OF CLASS A

[CHAP. V

From this and from (5.35) and (5.36) it follows that Inlz(re~)1 .

. I1m

=0.

r

r-.oo rEE

Comparing this with (5.34), we establish the lemma. REMARK. If an angle 0 < 01 ..;;;; () ..;;;; ()z < 1r does not contain points of the set {at} then within this angle the following limit exists in the ordinary sense:

lim In 11. (re(li) I

,,-,00

(a l < a < a2)·

=0

r

Indeed, from (5.34) (where one can obviously replace X(z) by Xl(Z» and from > 0 sufficiently small

(S.36) it follows that for 15

lim In I 1.1 (re") I r

= o.

The function Xz(z) is of completely regular growth with has no zeros. For ()I ..;;;; () ..;;;; ()z the limit

lim In lX2 (re la ) I r

~xponent

p=

I and

=0

exists, and hence lim In 11. (re"') I r

= o.

The next theorem yields a representation of special functions which are holomorphic in a half-plane. THEOREM S. If the function F(z) is holomorphic and of exponential type in the half-plane 1m z ;> 0, and if the integral 00

f

•

(5.38)

In 1F (I) I df 1+(.1

-00

exist.v, then 00

In I F(z) I =

f

fin 1F(t)1 -00

(1-:: + 2

yll+ky+ln Ix(z) ,.

(5.39)

where k = h'(1r/2) and X(z)=

Ii(I-~)(I-: )-1, a" a"

(5.32)

k-I

the at being the zeros of the function F(z) in the. half-plane y 11

> 0.111

R. Nevanlinna [2J established this proposition under somewhat different hypotheses.

Sec. 3]

FUNCTION OF EXPONENTIAL TYPE

241

PROOF. First of all we note that the product x{~) on the right hand side of (5.39) exists. Indeed, the convergence of the integraf(5.38) implies by Theorem 1 that F(z) is of class A and by Theorem 4 that the product x(z) converges in every finite closed region of the upper half-plane free of points at. Let us now construct the function (5.40) By Theorem 4 and the remark at the end of §2, this function is harmonic in the upper half-plane and takes on the value u(x) = lnIF(x)1 on the real axis. Let v(z) be the harmonic function (when y > 0) which is the conjugate of u(z). The function (5.41) ~(z)= F(z)e-" (.)-i., (1) is obviously holomorphic in the upper half-plane, and its modulus has the limiting value one on the real axis. By the principle of symmetry, this function can be extended over the lower half-plane, and it has poles at the points at of the same multiplicity as the zeros at of the function F(z). In order to obtain an estimate of the function Itp{z) I in the upper half-plane, we note that by Lemma 5

Iu (z) -- ky/ < c. + e ~ • Since F(z) is by hypothesis of exponential type,

In 1~(z)1 < c.+a~ +(a+e)r.

(5.42)

Finally, it follows from (5.41) and from the equation k - hz;l.."./2) that h"(.,,./2) <; O. In order to sharpen the inequality (5.42) we shall apply the procedure used in the proof of Tbeorem 4. We introduce the averaged (mean) function -It (z)

= 4!1

w+. '1:+"

f f

,-It

Iq I~(~+ It) Idt d't,

,-It

This is a subharmonic function when y > 0 and satisfies the conditions I) _.(x) = o. 2) _" (ly)

= 0 (y).

3) w" (z)

= 0 (r2).

Applying the Phragmen-Lindelot theorem to the function wA(z) - ey in each of the angles 0 <; arg z <; ."./2• ."./2 <; arg z <; ."., we find that for every

242

FUNCTIONS OF CLASS A

[CHAP. V

> 0 the function wh(z) - ey is bounded in the half-plane y ;;> O. Again, applying the Phragmen-Lindelof theorem to this function in the half-plane y 0, we see from (1) that wh(z) - ey <; 0 for every E > 0, and y > O. Since E > 0 is arbitrary, it follows that wh (z) -< 0 (y> 0). E

>

Taking the limit as h - 0, we obtain In "Hz) I -(: 0 namely,

(y>O).

I~(z) 1-< 1

>

in the half-plane y o. We shall next show that the function Y'(z) is equal to the product x{z) multiplied by a constant. For this purpose we set XII (z) =

IT (1 - a:) (1 - a:)

-1 •

k=l

The function is bounded in the upper half-plane, and since IY'n(X) I = 1 on the real axis, we conclude from the Phragmen-Lindelof theorem that lY'n(z)I ;s::;; 1 when 1m z ;;> O. Taking the limit as n - 00 and setting Y' oo(z) = Y'(z)X-1(z) we obtain

I ~",(z) 1-< 1

(1m z

> 0).

This function has no zeros or poles and hence !p (z)

= - t In ~'" (z)

is also a real entire function for 1m z > 0, 1m tp(z) > O. By the lemma of N. G. Cebotarev, tp(z) = rJ.Z + p (<< ;;> 0, and 1m p = 0). To determine « we select a sequence y; - + 00 such that . In I F(iYj) I 11m - - - = k .

Ili~+'"

Yj

From the inequality 1m tp(iy) <; u(iy) - In IF(iy) I it follows that

This implies that « = 0 and Y'c,,'{z) = eifj or Y'(z) = eifjx(z). To complete the proof of the theorem, one needs only to substitute this expression into (5.41).

Sec. 4]

FUNCTIONS OF COMPLETELY REGULAR GROWTH

243

4. FunctioDS of Class A aod of Completely Regular Growth

On the basis of the representation (5.39), we shall obtain in this section a number of theorems which yield useful criteria to determine whether a function of exponential type is a function of class A and of completely regular growth. It is useful to note that the density (when per) z:: I) of the set of zeros oran entire function of class A is equal to zero within any angle of the form 0 < 15 <; larg zl ~ 1T - 15 (see (5.37». Therefore, for every entire function of exponential type, of class A, and of completely regular growth, the density a('P) of the set of zeros is a piece-wise constant function which has only two jumps of equal height at the points 0 and 1T. From this it follows, that the indicator diagram of such a function is an interval (straight line segment) parallel to the imaginary axis. THEOREM 6. If F(z) is a function of exponential type in the half-plane 1m z ;> 0, and if the integral CJO

f

In I F (t) I dt

•

1

-ex>

+t3

(5.43)

converges, then F(z) is a function of class A, and of completely regular growth (0 < arg z < 1T), whereby h F«() = k sin (). PROOF. Under the hypotheses of the theorem the function In IF(z)1 admits the representation (5.39) in the half-plane 1m z ;> O. By Lemma 3, the first summand on the right hand side of (5.39) (after division by r) tends to zero when r - 00 and I() - 1T/21 ~ 1T/2 - 15 (d > 0). From Lemma 5 we obtain

lim In Ix (re i8 ) I = 0, r ,. .... ex>

,."fE where E is some EO-set, and therefore '8

InIF(re')1 . \ 1m

r

'6 = kSID

(5.44)

When 15 ~ () <; 1T - 15. Since d > 0 is arbitrary, it follows that for some set E of zero relative measure equation (5.44) is valid within the entire angle 0< () < 1T (Chapter III, Theorem I). THEOREM 7 (M. CARTWRIGHT [1]). If F(z) is a function of exponential type in the half-plane 1m z ;> 0, and if the integral ex>

r In

•

-ex>

+ I F (I)

1 +13

I dt

(5.45)

244

FUNCTIONS OF CLASS A

[CHAP. V

converges, then F{z) is of class A and of completely regular growth when 0 arg z < Tr, and hF{O) = k sin 0 (0 < 0 < Tr).

<

PROOF. We shall prove that for a function of exponential type, the convergence of the integral (5.45) implies the convergence of the integral (5.43). Then the assertion of Theorem 7 will follow directly from Theorem 6. For this purpose we make use of the formula of Carleman, which can be written in the form R

f (1°_IF

~

F(-/)IJ (~ - ~,)dl

(t) 1+ 10_1

).

+ ~ (A-~~)sin6k=2~ ).<·"k<;R

R

J

(lo+IF(t)I+ln+IF(-t)U

).

(t~ - ~2) dt+ 1t~

"

J

+

In 1F (Re i8 ) I sin II dO A). (F. R). o For a function of exponential type, the sum of the last two terms is bounded from above. We note also that X

~ (~- ~)sin6k~O. ).
and obtain B

f (1°_1

F (I) 1+ In_I F (-I) 11

(~- ~3) dt

).

R

<

f

[In+1 F(t)I+ln+ IF(t)/1

(~ - ~)dt+O (1).

).

Next, from this ancl from the obvious inequality x

"2

f

(In_I F (t)

1+'In_1 F (-t) I}:

).

J.

<:

f

[In_I F{t)l+ In_IF{-t)1J

(~- ~)dt

).

it follows that R

2"

J[In_IF{t)I+ln_IF(-t)lJ~ ).

R

<:

J

(In+IF(t)I+ln+IF(-t)IJ: +0(1)

).

Sec. 4]

245

FUNCTIONS OF COMPLETELY REGULAR GROWTH

and finally B

'2

+ In_I F (-t) 1dt SIn_I F(t) 11+t3

).

B

f

< ~.

In+IF(t)I+ln+IF(-t)ldt+O(l) 1+t3 I

).

where Cl is a constant. Under the hypotheses of the theorem, the last inequality implies 00

F (I) II dt < f lIn 1I+t3 00, -00

i.e., the absolute convergence of the integral (5.43). COROLLARY. The functions of exponential type in the 1uJ1f-plane 1m z :> 0 that are bounded on the real axis (or belong to £1'(-00, 00» are functions of class A, and are of completely regular growth in this half-plane.

Indeed, such functions obviously satisfy the conditions of the theorem of Cartwright. 13 THEOREM

1m z

8.

If F(z) is holomorphic and of exponential type in the IuJlf-plane

:> 0, and if it satisfies the conditions 00

SIn I F o

(x) F (- x) 1 Xli

+

I dx

exists

hp (0)+ hp (1t)

= 0,

((J)

then the function F(z) is of class A and of completely regular growth in the upper IuJlf-plane l4. (0 < (J < 17).

<

PROOF. Let us set F(z) == F(z) (1m z 0). The function fP{z) == F(z)F(-z) satisfies the hypotheses of Theorem 6, and hence '8

h, (6) = lim-In I, (re' ) 1= 2k sin 9, r-+oo

(5.46)

r

where k == h~1712). On the other hand, from condition (fJ) it follows that tbe indicator diagram of the function F(z) is an interval parallel to the y-axis. Without loss of generality, one may assume that this interval lies on the y-axis

<

11 For a function of L~ - 00, 00), this follows from the inequality p In+ If(~)1 I/(~)f •. U A theorem similar to Theorem 8 was proved by N. Levinson for entire functions of exponential type (see [1], 25-42).

246

FUNCTIONS OF CLASS A

[CHAP. V

(it is sufficient to multiply the function by e'" where y is some real number). Thus hp(f:I)= lim InIF(rei8"=ksinf:l. (5.47)

,,"'00

r

From (5.46) and (5.47) we have lim In I F(re i8 )1_ r

_

lim In I" (re i8 ) I ,. ... 00

lim

r

rEB"

(0

I IF( i IK-8»I n re =ksinf:l

,.-+00

,."E Eu

r

< 0 < 'It).

From this and from (5.47) it follows that . • In I F(re,ij)l k· IJ \1m = smu. r ,.-+00 This completes the proof of the theorem. 9. 15 In order that an entire function fez) of exponential type be of class A and that at the same time the density THEOREM

.

nr(r)

,.-+00

r

6., = 11m -

(5.48)

exist, it is necessary and sufficient that the integral 00

j' ~(x)f(-x)ld J +x~ x

(5.49)

o

exist. PROOF.

If the integral (5.49) exists then, by Theorem 6, the function If (z) = f(z)f(- z)

is of completely regular growth in the half-planes 0 < arg z < 71", and -71" < arg z < 0, and by the corollary of Theorem I, Chapter III, this function is of completely regular growth in the entire plane. The set of zeros of a function of completely regular growth has an angular density, and therefore the limit

,. \. n, (r) 2u,,= Im-,,"'00 r exists. For the establishment of (5.48) it is sufficient to note that n.,(r) = 2n,(r). Furthermore, the existence of the integral (5.49) implies that the function fez) is of class A. This is a consequence·of the criterion for membership in class A formulated in Theorems 2 and 3. U

A theorem of A. Pftuger [5].

Sec. 4]

FUNCTIONS OF COMPLETELY REGULAR GROWTH

247

Let us next prove the necessity. From the fact that the entire function/(z) of exponential type belongs to class A, and from the condition (5.48), it follows that cp(z) = j(z)f( -z) is an entire, even function of exponential type and of class A. The set of zeros of this function has a density. We recall (Gee (5.37» that all zeros of a function of class A, except possibly for a set of zero density, lie within arbitrarily small angles containing the rays arg z = 0, arg z = 71'. Therefore the set of zeros of the even function cp(z) has an angular density. Thus the zeros of cp(z) are regularly distributed, and hence cp(z) is of completely regular growth. For the proof of the convergence of the integral (5.49) we make use of Carleman's formula, applying it to cp(z) in the upper half-plane. First of all, we examine the quantity

~ (~- ~) sin Ok

(0

< 6k < 'It),

l
where hk = 'kei8k are the zeros of cp(z) in the upper half-plane. From the inequality 1 ~ • 6 ~ sin 6k R~

~

rk SIn

k

<

r.
r;;-

~

rk>r.

it follows that for a sufficiently large '.

~2

~

r k sin 6k

< e.

r.
This implies that

. 6 = O. . R2 1 ~ rk sm 11m k

B-+oo

r,,
In this manner we obtain 1·1m

B-+oo

Irk) .

~ ( - - R2 sm 6k ~ l
=. -

~ 1m -. 1 Ihkl >).

~

(5.50)

To investigate the quantity

"

K1

f

In Irp (Re i8 ) Isin 6 d6

o

we make use of the fact that cp(z) is of completely regular growth. From Theorem 3 of Chapter IV it follows that

. -r n., (R) =21 f h.,(6)d6. B-+oo 2"

11m

21:0

Hence

248

FUNCTIONS OF CLASS A

[CHAP. V

and by Jensen's theorem

J 2..

tim ,. .... 00

48

[In I , (Re ) ' - h,

R

o

(6») d6 = O.

(5.51)

If one makes use of the asymptotic relation (Chapter I, Theorem 28) In /, ~ei8)1

< h, (6)+ 8

(e> 0),

then it follows from (5.51) that

f Ik 2c

lim r-+oo

lin 'f ReIl )/_ h, (6)1 d6 = O.

0

.

From this we obtain the equation

lim

R .... oo~

Rl

~

SIn IIf (Rei~) I sin 6 d/J =1. f h, (6) sin 6 d6. ~

o

(5.52)

n

Since the quantity Alrp, R) also has a limit when R -+and (5.52) that the limit

00,

it follows from (5.50)

R

f

RI~~ (x~ - ~) In IIf (x) Idx

(5.53)

).

exists. It remains to prove that the limit

lim

R

f

~

In If(x)f( -x)/dx

A

exists. We have R

R

~

~f X Inl'f(x)l d =[~f Inl'f(t2..1dt]R_...!...fJ~(O)d X X R3 t R~' x.

R3

A

).

A

A

By Lemma 2 of Chapter III the limit tim R-+oo

! J:

(0) = h, (0)

exists, and therefore

lim R-+oo

~

R

SIn IIf (x)1 dx =

);

~ h,(O).

(5.54)

Sec. 4]

249

FUNCTIONS OF COMPLETELY REGULAR GROWTH

From this, and from (5.53) it follows that the integral

f

GO

In II(X~( - x)1 dx

A

converges. In order to deduce from this the convergence of the integral (5.49), we set

f

:z:

4(x) =

In 1/(1)£( - t) 1dt.

A

Integrating by parts, we obtain B

B

f 9(X)(~)' r Inl/(x)/(-X)ldX=9(R)~_ + + + dx 1

o

RlI

Xli

1

1

A

Xli

•

and the existence ofthe limit when R -+ ao is obvious. This proves the theorem. IfI(z) is an entire function of exponential type and of class A, if its set of zeros has density, and if181(0) = I then 00

~ J' In I/(x)/( -x)1 dx, 1t

xli

•

o

(5.55) For the proof it is natural to start from formula (5.02'). From the equations 9l(0) = 0, h,,(O) = h,,(1T/2) sin 0 (0 < 0 < 1T), (5.52) and (5.54) we obtain the equation 00 (5.56) In I 'f (x)1 dx =! ~ 1m ~ - ~ h (~), .. Xli ~ bk 2' 2

:S

Imllk> 0

o

which yields directly (5.55). We note that in view of (5.55) the equation

f 00

•

In If(x)/( - x)ldx Xli

=_

~ Iim~ In 1/(;y)1

2

r-+ oo

+ In t/( -ly)1 y

o is equivalent to stating that all the zeros off{z) are real. I7 We note also that the 11

17

This condition obviously does not reduce the generality or the assertion. This is a theorem or Paley and Wiener (1).

[CHAP. V

FUNCTIONS OF CLASS A

250

existence of the integral on the left side of (5.55) for an entire functionf(z) of exponential type implies the convergence of the series and the existence of the limit on the right side of the equation. IS The inequality (5.55) can be transformed by noting that by Theorem 3 of Chapter III n~ (r) 1 lim - ' - =-2 r-+oo r 'It

S 2"

hlf(8)d8

o

= -'It2

hlf

(

'It ) "2 '

and hence

TZr (r) = -1 r 'It

lim - -

r-+co

( 'It) hlf -2 •

Substituting this into equation (5.56), we obtain

~ 'It

co

co

SIn 1!(x)!(-x)l dx = ~ ~IIm~I-~ 2 X2

o

ak

lim nr(r). r-+oo

r

1

Here, by Theorem 9, the convergence of the integral on the left side of the equation is a necessary and sufficient condition for the convergence of the series, and for the existence of the limit on the right hand side. THEOREM 10. 19 In order that an entire function f(z) of exponential type be of class A and of completely regular growth, it is necessary and sufficient that it satisfy the following conditions: co

f

o

Inl!(x)!(-x)ld 1 x3 X

+

.

eXISts

(oc)

(P) PROOF. The sufficiency of the conditions follows from Theorem 8 since complete regularity of growth when 0 < () < 1T and 0> () > -1T implies regularity of growth in the whole plane (Chapter III, Theorem 1). The necessity of the condition (oc) follows from Theorem 9 because the set of the zeros cf an entire function of completely regular growth has a density. The necessity of condition (P) is obvious since the indicator diagram of an entire function of exponential type, of class A, and of completely regular growth is an interval parallel to the imaginary axis. Let us now combine some of these results into one theorem.

The converse can also be proved. A theorem of A. Pfluger. Govorov [1; 1J studied functions f(z) that are holomorphic and of exponential type in the half-plane 1m z > 0 and showed that Pfluger's conditions (a) and (fJ) are necessary and sufficient for f(z) to be of completely regular growth in the closed half-plane. For f(z) to be of completely regular growth in the open half-plane, it is necessary and sufficient that (a) is satisfied and that the two limits 18

19

lim !J.'ln1f(x)1 the, '_00' 1 X

lim

!f -'lnlf(x)1 the x

r-+CIO'

_]

exist.

Sec. 4]

FUNCTIONS OF COMPLETELY REGULAR GROWTH

251

THEOREM 11. If an entire function of exponential type satisfies one of the fo /lowing conditions: 20

f <Xl

(a) the integral

In I[(x)[( -x)1

o

f

l+x

2

dx exists, and h,(O)

= h,(7T) = 0,

<Xl

(b) the integral

In 1[(x) 1 l+x

2

dx exists,

-<Xl

co

f

(c)

In+lf(x)1

1+x3

d·<x 00,

-co

(d) I/(x) I is bounded on the real axis, IJ(x)IELP(-oo, 00),

(e)

then (1) f(z) is of class A and is of completely regular growth, and its indicator diagram is an interval on the imaginary axis; (2) all the zeros of the function fez), except possibZv those of a set of zero density, lie inside arbitrarily small angles larg zl < E and larg z - 7T1 < E, where the density

~i = lim

(i

n't(t)

=

1, 2)

t-+co

of the set of zeros within each of these angles IS equal to dj27T, where d is the length of the indicator diagram. Furthermore, the limit c5 = Iimr-<Xl d(r) exists, where ~

a(r) =

a;;l

lanl
(3) the function j(z) can be represented in the form

fez)

=

czme iM lim

IT (1 - :)n

R -+ co I an I < R

(k is a real number). PROOF. Assertion (I) expresses essentially the assertions of Theorems 6,7, and 8. Assertion (2) follows from Theorems 3 and 4, § 3, Chapter III, on the 2. These conditions are obviously not independent. Each one (after the first) implies the preceding ones. Indeed, by Theorem 6, it follows from (b) that hjJ) = k sin 0 (0 < 0 < ".), namely, h(O) - h(7T) = O. In the proof of Theorem 7 we showed that (e) implies (b). It is obvious that (d) implies (c). It is not difficult to show also that (e) implies (d) (this follows directly from a theorem of S. Bern~teln, which we shall prove in Chapter IX). In an article by Boas [21 a detailed investigation is given of the relationships among various conditions of this type; see also Boas [II.

252

FUNCTIONS OF CLASS A

[CHAP. V

distribution of the zeros of a function of completely regular growth. Assertion (3) is new. For its proof we use the representation of an entire function of exponential type and of completely regular growth

f (z) =

II (I -;.),

zwa,oz+b lim

R-+oc lalll
II

which was given in the remark after Theorem 4 of Chapter III. We must show that the a occurring in this representation is purely imaginary. For this we note that the function cp(z)= lim

II

R-+oc I alll
(1-~) II

can be represented in the form /I

e

!p(z) = ll,(,.)-i,);

IT (1 - :) IT (1 -~)ea;

la"I<,.

k lale l ;;.,.

(Izl

= r).

Ie

Since lim,_oo II5(r) - 151 = 0, it follows from Lemma 5 of Chapter II that the indicator diagram of the function fez) can be obtained from the indicator diagram of 'P(Z) by a translation given by the vector d. Since this displacement does not move the diagram from the imaginary axis, Re d = O.

s.

Indicator Diagram of an Entire Function of Exponential Type, and of Class A

Let us examine in more detail the indicator diagram of a function of exponential type and of class A. It follows readily from Lemma 2 of this chapter that the indicator diagram of an entire function fez) of exponential type and of class A has an axis of symmetry parallel to the real axis.11 Indeed, if (5.57) then a>

~(Z)=e~THiP J[(l_'!')(I_!)-t, fez)

=

where" 2 1m a to deduce that

hi

ale

(5.58)

ale

+ 21:;'..1 1m l/a", fJ = 21m b, and from Lemma 5 it ih easy lime In I/(re")I-ln IT
(5.59)

r-+co

Therefore,

h,(ft) = hl( -

namely, the indicator diagram has the at

axiS

(5.60)

6) -1 sfn S,

of symmetry y =

This fact was first pointed out by M. G. Krein.

-,,/2.

Sec. 5]

253

INDICATOR DIAGRAM

Suppose that ell: = Re al:' We shall show that the indicator diagram of the function/(z) and the function 00

/1 (z) =

•

II (1 - f) ea;

(5.61)

Ie

kml

can be obtained from each other by means of a parallel shift along some vector. For this we note that if we set

then

fez)

It (z) =

e"H'Y.1 (z).

By an unessential variation of the argument used in the proof of Lemma 5 of this chapter one can show that22 Iim~ In I Xl (r~)1 = 0 •

r

,.-+00

(5.62)

and hence that (5.63) h, (6) = hfl (6)+ I « I cos (6 + 60) (6 0 = arg (I). If the entire function of exponential type and of class A has completely regular growth then, as was shown, the indicator diagram is an interval parallel to the imaginary axis. One might conjecture that this is a property of all entire functions of exponential type and of class A. We shall show that this is not so by constructing the following example. EXAMPLE. Let us select a sequence of positive numbers {b .. } (b l ba < . . .) of density 6., that is,

< bz <

lim bn =11.,

n-+oo n

and a sequence of real numbers {cn} (Icll and such that the series 00

..; lezl < ICal < ...) of zero density

~,L, 1 diverges. (For example, one can take b.. = nand e.. = ±n In n.) We then select the signs of the Cft so that the limits of indeterminacy of the quantity (le(r) =

are equal to the numbers a and b (a II

Noble [I].

~ 1 en I en I <,.

< b).

254

FUNCTIONS OF CLASS A

[CHAP. V

Let us rearrange the sets of numbers {±bn} and {en} in the order of ascending moduli and denote them by {an}. The set of points {an} has density ~ on the positive and on the negative half of the real axis. Furthermore, for the quantity

~

oCr) =

I

4.J

I (I" I < ..

an

the following equations are valid: lim

oCr) = a,

oCr) =

lim

r .... 00

b.

(5.64)

.. -+00

The canonical product 00

/(Z) =

•

II(t - :n}ea;; nat

is an entire function of class A. By the theorem of LindelOf (Chapter I, § 14) it is of exponential type. Let us express fez) in the form

=

fez)

eO(r). /r(z),

where

We find that In I I (rete) I r

= 0 (r) cos (j

I_

I

From Lemma 5, Chapter II, with ()

~

0,

lim In IIf' (rei~) I r

7T,

=

In I Ir (re,1j) I r •

we deduce '1T~lsin 8 I.

Bearing in mind (5.64), we obtain '1T~lsin

81 + b LOS ~

hr (~) = '1T~lsin

81+ a cos 0

hI (~)

=

(1 0 1< ;), (10- 1<;). 1t

Thus, the indicator diagram of this function is the rectangle a '" x '" b, - Ll '" y :E;; ~. If we change the signs at all the points {en}, we obtain a function fl(z) which has for its indicator diaglllm the rectangle -b :E;; x '" -a, - ~ :E;; y :E;;~. The product h(z)f(z) is a function of completely regular growth with the indicator h«() = 2'1T~lsin 8 I. We have thus constructed an example of functions whose indicators are not added when the functions are multiplied. In this example the indicator diagram of the function fez) was broadened in the direction of the real axis because of the oscillation of the quantity b(r).

Sec. 5]

255

INDICATOR DIAGRAM

One can, however, construct an even function of exponential type (for which «5(r) = 0) with'real zeros which has an indicator diagram ihat is not an interval. The set of zeros of such a function obviously can not have a density, since otherwise it would be of regular growth and its indicator diagram wo Jld be an interval. Hence, the indicator diagram is broadened when the distribution of the zeros ceases to be regular. However, in this case the indicator diagram can not be arbitrarily wide in the direction of the real axis. The exact result is given in the following theorem. THEOREM 12. The indicator diagram of an even entire function f(z) of exponential type and of class A is situated, except for a parallel shift, within a convex region K with the supporting function

~ cos fJ· In(1

+ 2 cos fJV2 cos 2fJ + 2 cos 2fJ)

+ ~ sm fJ[ arctg 1 A

cos fJV2 cos 2fJ sin fJV2 cos 2fJ

•

k(6)=

11:

1 + cos fJV2 cos 2fJ + arctg--:--::-:--;:;:==:::::;:;;--

when 16 1< '4 and IfJ - ",1

1

sin fJV2 cos 2fJ

",

< '4' (5.65)

Here - - nr(r)

(5.66)

fl= lim - - . r-+oo

r

The region K is symmetric with respect 10 the y-axiS (Fig. 8). The inequality h,(O) <; k(O) is sharp since there exists an even entire function f(z) offinite degree and of class A whose indicator diagram coincides with K. II

Fig. 8 PROOF.

From (5.63) it follows that it is sufficient to consider only even

ell tire functions with real zeros, i.e" co

I(z) =

ll(l- ::) n- 1

n

(1m aft

= 0).

(5.67)

256

FUNCTIONS OF CLASS A

We have

J

[CHAP. V

<XI

In I f(re't) / =

r~2ia / dn(t).

In /1 -

n

The integrand In II - r2e2i8/121 takes on negative values when I> r(2 cos 28)-1/11 (181 < 7T/4). Therefore t

In/ 1(,,,,>/ <

'~j'" -, In 11- =,,;old.(/)

o Integration by parts yields t

r(200.')-'2

In If(re il) /

<-

f :,

II - ~ ellial} ;"(t)dt

{In

o

The integrand is negative on the entire interval of integration. Furthermore, < (L\ + £)1 asymptotically. Thus we have the asymptotic inequality

n(1)

t

r,2 C08 28)-'2

In / f(re i ,) /

f

< (~+ £)

~

{-In

o

II - ~ t2itll

t dt

(1 0 /< ~)

and hence

I

In /I(re,,>/ < (A+.) I!e ·~j·I)-'I. (I -

~ ....)tU

o

(/8/< :). Finally, setting t = Tr, we obtain t

In

1f~"')1 «A+,)Re

(t-"-'

J

In(I-~d~

o

(181~i). When 16 ± 7T/21

<: 7T/4, the function Inll-ra:

I

(5.68)

Sec.5J

257

INDICATOR DIAGRAM

is positive, and it is monotone decreasing when t is positive. Therefore, if in the preceding integrals we replace the upper limit by infinity, we obtain the asymptotic inequality

"" In If~e") I < (a +1) Re fin (1 _ ':) do; o (5.69) Evaluating the integrals (5.68) and (5.69), and making use of the arbitrariness of t:, we deduce that h,(O) k(O). This proves the first part of the theorem. Let us now construct an example of an even entire function of exponential type with real zeros and with an indicator equal to k(O). For this we select a sequence of positive numbers PI < P'I. < P3 < ... such that

<

tim p,,+t R+CO

p"

= 00,

and construct the infinite product

where the 01: are integers which lie in the intervals (P'l.tl-l' pl(I). It is obvious that -

n(t)

11m -t- == 1.

t+ co

For the determination of the indicator of the function f(z) when we select a sequence of values,r I

Til

= P2g-t (2 cos 28f'l.

It is easy to see thatla as q -

00

co

:q f In 11 1''1.1/

and

pr- In T I

to

II

1

I

11 -

r::

Idn(t'-+O

~e~

~ldn(t)-+k(fJ)

'Ibis is easily established if ODe iDfe&ra- by puts and replaces t by .".....

161 < ."./4,

258

FUNCTIONS OF CLASS A

When 77'/4 <; 101 <: 77'/2, one must take '
[CHAP. V

+ P2<1-1)/2.

Then it is ea"ily

r ll

or

f I 00

Ill:,:

In / , (rqelQ )

rq

/

""-> .,y

In 1 -

I . Ui I du = r.1 Sin fJ I.

e3l'l

o Combining this with the inequality h
=

k

(6)

(0

-< f.I < 21t).

REMARK. The function 'Pl(Z) constructed in the same manner as
6. Theorem of M. G. Krein on the Expansion of the Reciprocal of an Entire Function An interesting subclass of the set of entire functions was introduced by M. G. KreIn [2; 3] in connection with his development of the theory of entire operators in a Hilbert space. We are speaking here of entire functions fez) which admit an expansion of the type 1

fez)

=

R(z)

~ 1 [ 1 +_1 +-=-+ -l- ~p-I 1 + It=l ~ I' (all) z - all all a~ . •. at I

(5.70)

.J'

where R(z) is a polynomial and ao

~

1

~o I Co'le 11'+1/1' (all) I

<00.

The fundamental theorem of M. G. Krein is as follows.

(5.71)

Sec. 6]

259

THEOREM OF M. G. KRFiN

THEOREM 13. If an entire function f(z) of class A admits an expansion of the type (5.70), then (I) f(z) is afunction of exponential type and

f

00

(2)

-00

In+ I I(t) I dt 1 12

+

< 00.

Before proving the theorem, we note that (I) and (2) imply that f(z) is a function of completely regular growth and that its indicator diagram is an interval. 24 PROOF. We note that without loss of generality we can assume that R(z) == 0 and p = O. Indeed, it is easily seen that if the functionf-1(z) is expressed in the form (5.70) then, multiplying it by a polynomial P(z) of degree q > p, and q also greater than the degree of R(z), and such that its zeros are distinct from those off(z), we obtainfl(z) = Piz)j(z), which admits the expansion 00

1-1 (z) = ~

~

1

(5.72)

c"

z-a,,'

"=1 where the ak are all the zeros offl(Z), and where Lk=l ickl < 00. Let us next find an estimate of the functionf(z) which admits the representation (5.72) in the half-plane 1m z ~ 'YJ > O. For this purpose we circumscribe around each ak as center a circle of radius dk = (1/16A) 11m l/okllIm 0 10 1, where A = L k=l 11m l/akl. Let us denote by C(x, h/2) the circle Iz - x - ihl < h/2 when x and h are fixed, and by C the set of all exceptional circles. The sum of the radii of the circles corresponding to the zeros that lie in the circle C(x, h/2) satisfies the inequality 00

~

dk

<

1 16A

lak-w-ilal<.!

~ h ~ 11m a~ I < ~ h.

(5.73)

k=1

2

From this it follows that if the point x + ih lies in a region covered by the exceptional circles, then the entire region lies in the circle C(x, h/2). For every point z of the circle C(x, h/4) that does not lie in the exceptional circle we have the inequality

00

<. : ~ I Ck I+ "=1

~ I

,.

Ic" Idi 1

a,,-m-''', < 2'

I. This theorem was used by M. G. Krein [4] in his study of the spectrum of the SturmLiouville boundary value problem. Further generalizations of this theorem are given in Ostrovskii [1-3], Macaev [IJ and Sergienko [1; lJ. See also Ool'dberg and Ostrovskii [1], Chapter VI.

260

FUNCTIONS OF CLASS A

[CHAP. V

If one now takes into account the inequality dk - 1 <; SA that for every z E C the following inequality holds:

If-I (z) I <.

ft (I z P+ 1)

lak l2 41h 2, then one sees

h 3h) ( 2
(5.74)

where g is a constant that depends onf(z). Let us define the function x(z) by the equation co

X(z)=

11(1 _~)(I __Z)-1 a"

k=l

(1m ak

a"

> 0).

In the upper half-plane Ix(z) I <; 1, and from (5.74) one can obtain, by the principle of the maximum, the inequality

-g_(!ZI2+ I) Ilmzr

If-1(z)x(z)1 <

.

(1m

z > 0).

(5.75)

From this it follows that if 't{z) = j-l(Z)X(Z), then the function In 1q;(z)1 can be represented by a Poisson integral (see Theorem 4) in the half-plane 1m z ~ 'rJ > 0; namely, co

Inl'P(Z+ilj)l=l. n

f

(Y-1) ln lcp(/+llj)1 dt+k(y-"l) (I - x)3 (y - lj)3 '

+

-00

(5.76)

where k

=

lim

In I cp (ly) I

Y"+OO

Y

•

By (5.75) we find that

f

00

In+ I

i ~!

11)

I dt

< 21tg In ~ + gl (gl is a constant).

-co

Substituting z = x

+ iT} into (5.76), we obtain

- f Inl~(~-+;'llj)ldt=1tk-1tlnl'P(2lj+ 00

l)iJI

-00

or

00

00

f

In+ I , (t + l'l) I dl l+tlI

-00

+f -co

In+lcp-l(t+ilj)1 dt l+tlI

= 1tk - 1 t In ! ('11 (2lj

Combining this with (5.71), we find that for 0 00

f

+ 1) iJ I.

< T} <; 2

In+lf(t+Il)x-l(t+ilj)ldt <2 I 1 1 +13 P n lJ

-00

(gs is a constant),

+ g'},

(5.77)

Sec. 6]

THEOREM OF M. G. KREIN

and bearing in mind that

IX-1(1

+ i1)1 ;;.

261

1 we see that

00

f

In+ If(t+lYj)1 dt I III

+

<. 2g1t In ~+ Yj g2·

(5.78)

-co

An analogous inequality holds also when """:2 <;; 1) < O. Integrating the terms of the inequality (5.78) with respect to 17 from -2 to 2, we obtain 2

co

f f -2

ill) I ++ fA

In + 1f (I

I

(g .

< gs

dt dTj

3 IS

) a constant.

(5.79)

-co

Making use of the theorem of Jensen for the entire function/(z), we obtain ~lt

In I/(zo) 1

< 2~

f

In 1!(zo+re i8 ) 1dO.

o

If we multiply each term of this inequality by r and integrate the result with respect to r from zero to one, we obtain

Inl!(zo)/<'~

2" t

Jf

In/!(zo+rei8 )lrdrd lJ.

o u

Finally 2..

In+ I!(zo) 1<'

~

1

Jf

o

1n+ I!(zo+ re i8 ) I r dr dIJ

0

and from (5.79), with IYol <;; 1, we obtain the estimate In+ I!(zo) I

< ~ [l + (I xol + 1)'1) •

(5.80)

We shall prove thatf(z) is of exponential type. The inequality (5.80) gives an estimate for the function /(z) in the strip 11m zl <;; 1. In order to obtain an estimate of this function outside this strip, one can use the representation (5.76) by setting, for example, 1) = 1. Noting that If(z) 1 <;; Icp(z)I-1 when 1m z > 0, we can conclude, on the basis of the representation (5.76) and by Lemma 3, that when r > r. and lyl > 1 then Inl!(z)1

< klyl+e

r3 1 • /IYI- 2 /

(5.81)

It now follows that for Y ;;. 1 In I! (z) I =

0

(r 2 ).

(5.82)

262

FUNCTIONS OF CLASS A

[CHAP. V

Combining this with the inequality (5.80) we see that (5.82) is valid in the whole plane. By applying the Phragmen-Lindeloftheorem in each of the angles larg zl <; {) and larg z - 771 < {), we establish the inequality

111 (z) I < Mte(k+a),. where Ml is a constant. Therefore.f(z) is of exponential type in the whole plane. In order to obtain the second assertion of the theorem. we note that In+ 1/(1 + z)1 is a subharmonic function 25 of z. From this it follows that the function 8

ITr

or8

S In+ 13+ If(t+z)1

()_

z -

1

dt

-8

is also subharmonic. As a direct consequence of the definition of a subharmonic function we obtain the inequality 21<

q'R (0)

J

<: ~

11"R (e i8 )

db .

o

The estimate (5.78) now yields R

j'

In+ 1/(/) 1 dt 1 tA

+

< 4g'lt In 2+g;a.

-8

from which it follows that co

f

In+ 1/(1) 1 dt

1+/9

<

00.

-00

This proves the theorem. As was indicated earlier. conditions (1) and (2) imply thatf(z) is of completely· regular growth and that its indicator diagram is an interval on the imaginary axis. But one can say more than this about the growth of functions of class A which admit the representation (5.72). We shall confine ourselves to the case when all the a" are real. Indeed. the indicator of an entire function of exponential type

I.

A subharmonic function is a real function u(z) which satisfies the inequality

U

for all z aDd for R

> O.

(z)

'2~

f

2"

o

u (z

+ Rei') dB

Sec. 6]

THEOREM OF M. G. KREIN

263

is zero when 0 ~ 0 <; fT. From the complete regularity of the growth of the function/(z) it follows that the indicator of a function ,(z) == 1-1 (z) e- ika •

which is holomorphic and of exponential type in the half-plane 1m z also zero. From the representation (5.72) we can deduce directly that

I
kh

<;- ~ I <..0

cn I

= "'I

(h

> 0,

is

> 0).

1

This implies, by the Phragmen-Lindelof theorem for the half-plane, that if 1m z ;> h > 0, then Itp(z)!

<M

and hence26

II (z) I> MeJar ( ; )11 ).

(5.83)

An analogous estimate applies in the lower half-plane. It is interesting to note that if an entire function which is of exponential type and has its indicator diagram [-i,u, ill] on the ima~inary axis satisfies the inequality I/(z) I > C I z leI'- 111 I (5.84) (c 0)

>

in a sequence of circumferences, then it admits an expansion of the type (S.72). This follows easily from Cauchy's method for the expansion of meromorphic functions. Thus, for entire functions/(z) with real zeros, the necessary condition (5.83) differs little from the sufficient condition (5.84) for the expansion of the function ll!(z) into a series of the type (5.72).

:. This inequ.lity ..... s est.blished by M. G. Krein [1).

VI

CHAPTER

ROOTS OF EXPONENTIAL SUMS This chapter consists of two parts. In the first four sections we consider the distribution of roots for holomorphic almost-periodic functions. In particular, in §§ 2 and 4 we study the distribution of roots for almost-periodic functions with semibounded and with bounded spectrum. In the second part. consisting of §§ 5. 6 and 7, we il\vestigate the distribution of roots for entire functions of exponential type, which are approximated in a certain sense by finite sums of the forrr n

p (z) = ~ a"el.1!

(6.00)

1:=1

with complex exponents. The almost-periodic functions with bounded spectrum constitute a very special form of such entire functions. The investigation of the distribution of the roots for these functions is based on the results of the first four sections of this chapter and on the theorems and methods presented in the preceding chapters. The fundamental result is the theorem according to which an entire function fez), represented by the absolutely convergent series 00

fez)

= 1: a"el.,,·

(6.01)

"~t

in which the Air are points of the boundary of some bounded convex region 1, is a function of completely regular growth. Thus, by Theorem 3 of Chapter III there is an asymptotic formula for the distribution of the roots of these functions. Analogous theorems are proved for functions of more general form. Finite sums of the type (6.00) will be called exponential polynomials. 1. Infonnation from the Theory of Almost-Periodic Functions We shall need certain basic definitions and theorems from the theory of almost-periodic functions. We recall them for the reader's convenience. l 1

A detailed exposition of the theory of almost-periodic functions is given in the book of

B. M. Levitan [1]. Sec also H. Bohr [I].

264

Sec. 1]

THE THEORY OF ALMOST-PERIODIC FUNCTIONS

265

A set {T} of real numbers is said to be relatil'ely dense if any interval whose length is not less than some positive number I contains at least one number of the set. An interval of length I is called an enclosing intert·al of the set. The number T is called an E-shift of the function f(x) if for all values of the variable x the inequality

<

(6.02) 1/(x+'t)-/(x)1 E. is satisfied. The functionf(x) is taken to be complex-valued but defined for real values of x. The functionf(x) is said to be almost-periodic ifit is continuous on the entire real axis, and if to every real number E there corresponds a relatively dense set of E-shifts. We shall enumerate a number of properties of almost-periodic functions. (a) A necessary and sufficient condition for a function I(x) to be almostperiodic is the compactness on the whole axis of the functionsfh(x) = f(x + h), depending on the arbitrary real parameter h (S. Bochner). A set is called compact if every infinite subset contains a uniformly convergent sequence. (b) An almost-periodic function is bounded and uniformly continuous on the entire numerical axis. The sum and product and also the limit of a uniformly convergent sequence of almost-periodic functions is an almost-periodic function. (c) A generalized trigonometric polynomial, i.e., a function of the form n

p (x) = ~ a"eil.,p "=0

(6.03)

C)'A- a real number), is an almost-periodic function, since it is the sum of periodic functions. The uniform limit of such polynomials is also an almost-periodic function. (d) For each almost-periodic function there exists a "mean," i.e., T+&

M (f)

=

lim T .. oo

;1 f

I(t)dt,

(6.04)

-2'+&

where the convergence is uniform relative to Qt. The "Fourier coefficients" a (A) = M {/(x) ,-h}

(6.05)

differ from zero only for a countable set of values A. This countable set A., As, ... , A", ... is called the "spectrum of the functionf(x)" or the set of its Fourier exponents, and the series (6.06)

is the Fourier series of the function j(x).

266

ROOTS OF EXPONENTIAL SUMS

[CHAP. VI

Parseval's equation holds: .~)

(6.07)

This is equivalent to the uniqueness theorem. i.e .. the theorem according to which different almost-periodic functions have different Fourier series. (e) There exists a sequence ofmultipliersbyd(k = 1.2, .... n. 0 ~ W) < 1) and bkn ) --+ I as n .-~ 'X:,uch that for any E > 0 and n > N(E)

I I(x) -- ~ b~n)akeilka; 1< e

,,= t

(-

r:;o

< X -< 00).

(6.08)

It follows from this fundamental theorem that the class of almost-periodic functions is the closure of the class of generalized trigollometric polynomials. (f) If the sequence of generalized trigonometric polynomials n

P n (x) = ~ a~n)eilJi1' k=1

converges uniformly to j(x). then 01:.) --+ Ok' the corresponding Fourier coefficient of the functionj(x). A functionj(z), holomorphic in a closed strip IX ~ Y ~ fl (y = 1m z) is said to be almost-periodic in that strip if to each positive number E there corresponds a relatively dense !let of H,hifts of the function fez). i.e.,

If(z+'t)-/(z)l<e

for

cx
(6.09)

The functionj(z) is said to be almost-periodic in the open strip - co ~ IX < if it is almost-periodic in every closed strip lying in the strip For almost-periodic functions in a strip we have the theorem of S. Bochner, i.e., (a') the almost periodicity of the holomorphic function j(z) in the strip IX ~ Y ~ fJ means that the set of functionsj(z + h) for real h is compact in the sense of uniform convergence in that strip. -Moreover. j
< fl ~ 00 IX < y < fl.

Y

THEOREM 1. Suppose that j(x) is an almost-periodic junction defined on the real axis. For there to exist 0 holomorphic function of exponential type fl. in the half-plane 1m z > 0 which on the real axis has limit values coinciding with the I}aiues ofI(x), it is necessary and sufficient that the greatest lower bound of the spectrum ofI(x) be equal to - fl.. In addition I(z) will be almost-periodic in the ha/f-plane 1m z ;> O.

PROOF. Suppose that I(x) is an almost-periodic function and - p. Because of Theorem (e) there I"xists a sequence of polynomials fa

P n (x) =

~ a~n)eilttfD k=1

= inf At.

Sec. 1]

THE THEORY OF ALMOST-PERIODIC FUNCTIONS

267

which converge uniformly to the function/(x) on the real axis, while the numbers Ak are the Fourier exponents of the function lex). Since leiAtzl = rAtll <; e/AII (y = 1m z), the polynomials eip.zp..(z) are bounded in the half-plane 1m z > o. By the Phragmen-Lindelof principle, the moduli of these polynomials are bounded in the half-plane 1m z ;;> 0 by the same number as on the real axis. Thus from the inequality

/Pn(x)-Pm(x)l<e

(-oo<x
there follows the inequality

le;p.zPn(z) - e;p.zPm(z)1

<e

forlm z ;> 0,

and the sequence of polynomials ei/A'P..(z) converges uniformly in the halfplane 1m ? ;;> 0 to a bounded function ei/A' fez). The function f(z) is holomorphic I fez) I Mel'-Y for 1m z > 0 and

-<

(M

=

sup -00

<:z:
If(x) I.

y

= 1m z> 0).

(6.10)

i.e.,/(z) is of exponential type ~ ~ 1-'. Now suppose we are given that the almost-periodic function/(x) is a limiting function for the holomorphic function j{z) of exponential type ll. in the halfplane 1m z > O. This function is bounded on the real axis and accordingly, by the Phragmen-Lindelof principle, (6.11) If(z) 1-< Me AII • By Cauchy's theorem, the integral around the rectangular contour (- T, T, T + iy, - T + iy, - T) of the holomorphic function /(z)e iAz is equal to zero, i.e., •

21T

-I

-i

T

1·

f(x)e-iA:Z:dx- 2tr

+ 2~

f(x

+ ly)e-.A(II:+i,) dx

11

J

f(T+lt)e- i 1.(T+it)dt

o

r 11

- iT

f(-

o

T+ It)e-iA(-T+it) dt= o.

From inequality (6.11) it follows that as T zero and accordingly

00

(6.12)

the last two terms converge to

a(A)= M (/(x)e- i 1.:z:} = M {f(x +'y) e-i).(lI:+i,l}.

Hence, from inequality (6.11), it follows that

I a (A) I -< M~A+).)lI •

268

ROOTS OF EXPONENTIAL SUMS

[CHAP.

V[

If A < -~, then as y -. + co we obtain a(A) = O. Thus the spectrum of f(x) is bounded below and - p. = inf All: ;> -~. The theorem is proved. COROLLARY. For the almost-periodic function f(z) to be an entire function of exponential type it is necessary and sufficient that its spectrum be bound"d.

We observe further that, in addition, the smallest segment of the imaginary axis containing the whole spectrum of the function f(x) coincides with its conjugate diagram. 2. Roots of an Almost-Periodic Function with a Bounded Spectrum lfin the sum (6.01) all the exponents All: lie on the imaginary axis, so that the convex region I is a segment of the imaginary axis, then it follows from the corollary to Theorem I thatf(z) is an entire almost-periodic function ofexponential type. Thus in this case our problem comes down to that of the distribution of the roots of an entire almost-periodic function. The corresponding theorems will be presented in this and the following two sf!,tions. These theorems are needed 'not only when the convex region I is a segment of the imaginary axis, but also in the more general case when the boundary of I contains a rectilinear segment. We begin by proving some lemmas. LEMMA I. Suppose that fez) is a holomorphic almost-periodic function in the strip Iyl < h not identically equal to zero. Then to any number 15 > 0 there corresponds a number m(~) > 0 such that

Ifez) I>- m (3) for z lying in the strip Iyl < h - 15 and outside circles of radius 15 with centers at the roots of the function j(z). PROOF. Suppose that the greatest lower bound of the values of the function If(z)1 in the given region is equal to zero, i.e., that for some sequence {Zj} (j = 1,2, ... ), taken from this region, limj _ CXl f(z;) = O. By Bochner's theorem (a') one may select from the sequence of functions fIi;(Z) = f(z + Xl) (Z; = Xj + iy;) a sequence fIi",(Z) = fez + X",), which converges uniformly in the strip Iyl <; h - 15 to some function Io(z). Suppose that Z = iyo is a condensation point for the points iy",. From the condition

11m Cf'ni (IYn.) = lim f(x n ,

i-+a:.

J

1-+00

J

+ IYni ) = 0

it follows thatfo(iyo) = O. From the well-known Hurwitz theorem on the roots of uniformly convergent sequences of holomorphic functions, all the terms of the sequence flin(Z) from a certain point onward must have roots in a 15neighborhood of the point iyo. In other words, the functionf(z) has an infinite set of roots in the circles Iz - Zn1 I <15. But that is impossible, since the points

Sec. 2]

ROOTS OF AN ALMOST-PERIODIC FUNCTION

269

lie at a distance larger than lJ from the set of roots of the function f(z). The lemma is proved.

Z nj

LEMMA 2. The number of roots of a holomorphic and almosl-periodic function in the strip Iyl < h occurring in the rectangle I < x <; I + I, Iyl <; h - 6, is bounded by some number k(lJ) not depending on t.

PROOF. Suppose that there exists an infinite sequence of numbers t; such that the number of roots of the function f(z) in the corresponding rectangles increases without limit. We select from the sequence of functions f(z + ,;) a uniformly convergent subsequence. The limit function Io(z) must have an infinite set of roots in the rectangle 0 <; x <; I, Iyl <; h - 6, and thus it is identically equal to zero. But from the uniform convergence of the sequence of functions f(z + tTl;) to fo(z) it follows that

sup

Ifo(z)l=

-oo<~
i.e.,f(z)

== O.

sup

-oo<~
I/(z) I

(lyl
The lemma is proved.

LEMMA 3. If the spectrum of an almost-periodic function f(z) is bounded below and the grealest {ower bound - ~ enters into the speclrum. then

lim ei6(~+i")f(x+ly)=a(_d). 11-++ 00

while the convergence to the limit is uniform relative to x. PROOF.

Suppose that

"

P n (x) = ~ a{fI' (A,,) eo.lIfI' 1

is a sequence of almost-periodic polynomials, uniformly converging on the whole real axis to the functionf(x). In the proof of Theorem I it was shown that the sequence of almost-periodic polynomials Qn (z) = ei6 .pfI (z)

>

converges uniformly in the half-plane 1m z 0 to the function ei6 ·/(z). As y -+ + 00 the almost-periodic polynomial Q..(z) converges uniformly to its Fourier coefficient a(tI)( - ~). On the other hand,

lim a Cft) ( - d) = a(- d)

"-+ao

'

where a( -~) is the Fourier coefficient of the functionJtx). Choosing the number n so that \a(n)(-d)-a(-d)l< ;

and f!(x)-P" (x) 1<

;,

270

ROOTS OF EXPONENTIAL SUMS

[CHAP. VI

we find that for all sufficiently large values of y 1 ei4z

I (z) -

a (--

~) I <;;. 1 e1t.: II (z) - P n (z») ;

+1 eit.zPn (z) Thus, as y

-+

a(n)

+ 00 the function

(-~) I

+ I a(n) (-~) - a ( - ~)I < e.

I (z) eit.,

uniformly converges to a( - u). THEOREM 2 (H. BOHR). In order that all the roots {a~} of an almost-periodic function of exponential type u in the upper half-plane (1m z > 0) be situated below some line, parallel to the real axis (1m ak < h), it is necessary and sufficient that the greatest lower hound - ~ of the spectrum enter into the spectrum. PROOF. If - u enters into the spectrum, then a( - u) -:F 0, and by Lemma 3 the function f(::)e'j,z does not vanish for sufficiently large y (y ~ h). On the other hand, if all the roots of the function fez) lie below the line y = h, then the function In If(z)c,j,zl is harmonic and is bounded above in the half-plane y ~ hI (hI = h + r'J, b > 0) and accordingly, from Theorem 4 of Chapter V, it may be represented in the form In

I/eZ) ei4Z i

=

Y - hI 1:

~r

In

11/(t+ i1z 1 ) I e- 6h'l (t _

dt X)2

+ (y -

hl)~'

(6.13)

" _on

In this connection we make usc of the fact that, from Theorem chapter, lim ~lf(iY)1 =~,

of thIS

Y

for

By Lemma 1 there exists a number m > 0 such that In If(t -OCJ < t < 00, and from the equation (6.13) we find that I/(z)ei.!.zl

> me,,6h,

for

+ ihI ) I ~

In m

y>h 1 ,

Thus, lim/l_:X> f(z)eij,z -:F 0, so that the number -

u enters into the spectrum.

COROLLARY 1. If the lower bound of the spectrum of an almost-periodic function fez) is negatire and if it does not enter into the spectrum of the function, then fez) takes on erery mlue in the half-plane y > h > o. In fact, for any complex number A the greatest lower bound of the spectrum of the function fez) - A (the same as for fez»~, does not enter into the spectrum, and accordingly this function has a zero with arbitrarily large imaginary part. Corollary 1 is a well-known theorem of Bohr. COROLLAR Y 2. In order that all the roots of an entire almost-periodic function of exponential type be situated ill some strip parallel to the real axis, it is necessary and sufficient that the upper and lower bounds of the spectrum (the "ends") enter into the spectrum.

Sec. 2]

27\

ROOTS OF AN ALMOST-PERIODIC FUNCTION

We shall call such functions functions of class r11] if the width of the indicator diagram is equal to 211. A function of class (~] is, up to a multiplier of (he form eiAz , a function of degree 11. LEMMA 4. If the lower bound - 11 of the :.pectrum of the positiL'c function f(z) enters into that spectrum, then there t!xists a positive real number H such that if y > H the function argf(x + iy), considered as a function of x, may be represented in the form argf(x+ iy) = - .1x + 'P,(x). (13.14)

where CPI/(x) is an almost-periodic function. Here we have, of course, selected out of the infinite set of values of argf(x + iy) some continuous branch. Such a selection is possible since, for y > H,f(x + iy) ":F O. PROOF.

Represent the functionf(z) in the form !(z) = a (-.1) e-i<\z [I +~ (z»).

This is possible, since by definition a( - L1) ":F O. The Fouri~r exponents of the function ~(z) are all positive and accordingly, for sufficiently large H> 0 and y> H, will satisfy the inequality i~(x + iy)1 < 1. Taking a univalent branch of the function In [\ + ~(z)], for example putting larg [\ + ~(z)]1 < 7T for I,(z)1 < \, we find that b,(x)

=

In (\ +qx+iy»)

is a continuous function of the almost-periodic function 1 + ~(x + iy) ~nd, as follows directly from the definition, is itself almost-periodic. Thus we have

arg f(x+iy) = - t!..x+ 1m while cPy(x)

=

1m [Oix)

+ In a( -

[~,(x)+

In a (-~)l.

C.)] is an almosI-~efiodic function.

THEOREM 3. The roots of a function f{::) f, r :+1SS r~J lie in the strip Iyl and if n(t) is the number of roots in the rectangle Iyl ~ h, 0 < x ....;; t, then ~

n (t) = -11: t while the function w(t) is

+ w (t).

~

h,

(6.15)

bounded.~

PROOF. From Corollary 2 to Theorem 2, all of the roots of f{z) lie in the strip. Take H sufficiently large, and apply to the rectangle Iyl ..; H, 0 < x <; t,

• In a supplement we present the work of M. G. Krein and the author, in which we gave a complete characterization of the set of roots of a function of class (6).

272

[CHAP. VI

ROOTS OF EXPONENTIAL SUMS

the well-known formula for the number of roots of a function inside a contour. We obtain H

t

J' I' (u - IH) 1 rI' (t + Is) n (t) = 2ii I(u -IH) du + 2r. _~ l(t + IS) ds 1

0

t

i

H

+ lH) d _J... I' (IS) ds I(u + lH) u 2r. lOs) . -

__ 1 fl' (u 27t1

~

Here we are assuming that on the lines x = 0 and x = t there are no roots of the function f(z). The first of these assumptions may be made without destroying the generality of the considerations, since one may consider the functionf(z + E) instead of the functionf(z). Taking the real parts of both sides of the equation and making use of Lemma 4, we obtain

21rn (t)

= 2 ~t -!PH (t) + • -H (t) + Re (II

I

H

+Is) + c,

I' (t + Is) ds f(t

-H

(6.16)

where !Pll(t) and !P_H(t) are almost-periodic functions and H

rI' c= -Re -i

(Is) -/(15) ds

+!PH(O)-!P_H(O).

We shaIl prove the bounded ness of the function wP) = 2M(t)- 2 tJ+'II(t)-'_H(t)-C.

We shaIl caIl exceptional those values of t for which the line Re z == t intersects circles of radius 15 > 0 with centers at the roots at of the functionf(z) These values evidently form intervals, which we shaIl caIl exceptional intervals. The bounded ness of the function wet) outside the excluded intervals follows easily from its integral representation II

w(t) 1

= Re

f /' -II

In fact, from the formula

I' (z) == ~I

(t+ IS)d /(t+ls) S.

f (/ (t~)1 d:

IC-·I-l

there follows for Iyl <; H the inequality

1f'(z)1 < M,

in which

M=

sUP I/(z) I· Irl
Sec. 3]

HOLOMORPHIC ALMOST-PERIODIC FUNCTIONS

273

On the other hand, for I lying outside the exceptional intervals, from Lemma I + is) I ~ m(~) > 0, and accordingly

1/(1

I Olt(t)1 -< 2HMm -1 (~). In order to estimate the function (0(1) in the exceptional intervals we observe that by Lemma 2 their total length in any interval (x, x + I) does not exceed 2kd, where k is the largest number of roots of the function.f(z) in the rectangle t < x '" t + I, Iyl '" H. We shall suppose that ~ < 1/4k. Denote by Xes) the least upper bound of the oscillation of the function 2~t - IPH(t) + IP-H(t) on the interval (x, x + s) for - 00 < x < + 00. Because of the uniform continuity of the functions IPH(t) and IP-H(/), the function Xes) is bounded. For a given value of I we choose the number s(lsl < 1) such that the point t + s lies outside the exceptional intervals. Then

I Olp+s)- Ol1(t) I -< 21tk +X ({) and accordingly

IOl1(t) I < 2HMm-l(o) +

21tk+X ({)

for all values of t. The theorem is proved.

3. A General Theorem on Roots and on Mean Motion for Holomorphic Almost-Periodic Functions The mean motion for a complex-valued function I(x), defined on the real axis and different from zero, is the limit . argf(x)-argf(y) I1m c,= :1)-,,"'00 x - Y

Here we have chosen some continuous branch of the function argf(x). If the function I(x), holomorphic on the real axis, vanishes at certain points, one may distinguish "right" and "left" arguments of the function I(x) at these points. The right argument is defined as follows: at any point Xo one chooses arbitrarily a value of arg I(x) and continuously extends it to the right up to the nearest root of I(x). One goes around this root in a small semicircle below it and one again extends it continuously to the right along the real axis. In the same way one extends arg I(x) to the left, going around and below each root. Evidently on passing from left to right around a root of the function of multiplicity p the right argument is increased by prr. The left argument is obtained in the same way by going around the roots of the function in a semicircle above them. Evidently on going around a root of multiplicity p, the left argument is increased by -TTp. Correspondingly one defines right and left mean motions

274

ROOTS OF EXPONENTIAL SUMS

[CHAP. VI

ct

cj and c,. If = c,. then this quantity is called the mean motion and is also denoted by C,.3 In this section we shall present some of the theorems of Jessen on the mean motion and the distribution of the roots I)f holomorphic almost-periodic functions. LEMMA 5.

For el'ery almost-periodic function f(z) holomorphic in the strip

< y < tJ and not identically equal to zero. there exists a mean ~ (y) = ;W ( In I f(x + iy) I ) for ex < y < fl. IX

(6.17)

:II

+

PROOF. Put If(z)lm = max {If(z)l; m}. The function In If(x iy)lm evidently is an almost-periodic function of x and therefore has a mean

fPm(Y)

= ,W {lnlf(x+ly)lm}'

(6.18)

:l:

Construct the square (I): Ix - xoi < I and Iy - Yol < I. and denote by k the iargest possible number of roots in the intersection of thIs square with the I The first problem reiated to m'~~J :notion was po~d by Lagrange in 1782, in connection with an astronulI.ic2 1 !"roblem. It mav r:>e formulated as follows. Prove that every generalized tri~onom::tri~al piJI~·r.omilll has ~ mean motion. Many subtle investigations have ;'c:~'n devoted to this problem. Here we shall mention a few oi :hem. Lagrange himself consdered only the trivial case when the m:>duius of one of the coefficients exceeded the sum of the m0duli of all the others. The Riga mathematician Bohl .howed that a generalized trigonom,·t·:c polynomial consisting of three terms always has a mean motion. in 1961 H. Weyl showed that if in the generalized trigonometrical polynomial

Pfx)

n

')'

~ a~' lID

=

1e-1

II

the numbers A. were linearly indepe"dent (i.e., there are no integers m. such that! A.m. = 0 and 1.'

2:" m~ #

1'=1

0). then the polynomlai has a mean motion, Subsequently, developin~ his

~l

method. H. Weyl [1) proved that the :.agrange problem has a positive solution. However. the proof of Weyl contained a mistake. \;aking the necessary corrections, "essen and flJrllhave [1) gave in 1945 a complete solution 01' the question. In this paper Jessen and Tomhave also presented a detailed history of the qut:suon, with the fundamental res~:ts and with a complete bibliography. The problem of mean motion may be posed not only for generalized trigonometrical polynomials, but also for arbitrary almcst-penoaic functions. The following theorem or Bohr (1) is fundament .. l here. The ar.fumenl of an 'llmOSI-periodic -,"unction {(I). salisfying Ihe inequality! f(1)1 ;> m > 0 on Ihe entire numerical axis, may be "'.,,,,'stmtea in the form argJft) = ct

+ '1'<1).

where c is a conslanl and '1'<1) an almOSI-periodic function.

Sec. 3]

HOLOMORPHIC ALMOST-PERIODIC FUNCTIONS

275

strip IX + ~ ~ Y ~ f3 - b(~ > 0) for a fixed Yo and with Xo varying from - 00 to 00 (see Lemma 2). Choose a number 151 > 0 so small that the inequality 2k~1 < ! is satisfied. The set covered by circles of radius ~1 with centers at the roots of the function f(z) consists of regions. The diameter of each such region (I); (j = 1, 2, ... ,p) intersecting the square Iy - Yol ~ !, Ix - xol ~ l evidently does not exceed the quantity 2k~1 < ! and therefore each such region will lie in the square (/). Outside of these exceptional regions we have, from Lemma I,

If(z) I~ m(o,,) > 0 Consider the integral 1 11:.+"2

J=

f

[lnlf(x+ly)I",-lnlf(x+ly)I]dx 1

:1:.-'2

in which m = m(c5) and Iy - Yol ~ l. Outside the exceptional regions (j = 1,2, ... ,p) the integrand is equal to zero. Construct the polynomial

WI

q

P (z) =

II (z ---'- zs) •

.. =1

in which the z. are the roots of the function f(z) in the square (I) (q ~ k). Evidently the inequality IP(z)1 ~ (2V2)k holds throughout (/). On the boundary of the exceptional region W; we will have

If (z) p-l (z) I ~ m (2 V2) -k. By the maximum principle, this inequality will hold in the whole square Ix - xol ~ !, Iy - Yol ~ !, and accordingly 3

In If(x+iy)1 ~ In m- 2'k In 2+ln IP(X+iy)1

(lx-Xol<

~,

lZ+o
~-o),

and in the regions w; we will have the ineq uality q

In/f(x+ly)/",-lnlf(x+ly)1

< ; kln2- ~Inlz-zal. a-I

Since the diameter Wi does not exceed 2k~I' we therefore obtain 211:6.

1J1<3k28\1n2-k

f

-2ka.

This estimate holds for any Xo-

Inltldt

(lZ+a
276

[CHAP. VI

ROOTS OF EXPONENTIAL SUMS

Letting IJ tend to zero, and therefore also m(6) --+ 0, we find that J uniformly relative to 0: + b Y ~ {3 - 6. Writing

<

cp(y, xJ.'

--+

0

"'0

x~= xa~xl

f In /J(x+ly)1 dx 3:.

and constructing fPm(Y,

Xl' X2)

/~(y,

analogously from In /I(x

Xl' X 2)-CPm(y, Xl'

for m < m. and arbitrary Xl and x 2 (X2 > Xl Thus we obtain for sufficiently small m ~m (y) -

e

<

lim

+

+ iy)lm, we obtain

x\l)/<e

1).

cP (y, Xl' X2 )

X , -X1 -+CO

Therefore the existence. of If(y) follows. The lemma i~ proved. Let us study the function If'(y). To this end it is convenient to use the argument principle. Denoting by n(x1 , X 2 , s, t) the number of roots of the function fez) in the rectangle Xl < Re z ~ x 2 , S ~ 1m z ~ t, situated in the strip of almost-periodicity of fez), and applying the argument principle, we may write ~t

21tn ( Xl' X2'

XI

_Sdarg/(x+lt)d t) -_fdarg/(x+IS)d dx X dx x

S,

(6.19)

The last term in this equation, which we will call C(xl , x 2 , s, I), is a function bounded in the relatively dense set E of values of the variables Xl and x 2 (cf. the discussion in Theorem 3). From the Cauchy-Riemann conditions it follows that

d[ argf(x + is)] dx

d In

=-

I I(x + Is) I ds

and after substitution we obtain

21tn(x p x 2 '

S,

t) .2:.

:1:,

=_jdlnl/(x+IS)l dX ds

1/:.

+ •fdlnl/(x+lt)/dX+C(XI' x dt 1/:.

2'

s, t). (6.20)

The integrals on the right side of equation (6.20) exist for all values of the quantities s and I with the possible exception of a countable set.

Sec. 3]

277

HOLOMORPHIC ALMOST-PERIODIC FUNCTIONS

Let rl < YI < Y2 < Y3 < p. Integrating both sides of equation (6.20) in s between the limits YI and Y2' and in t between the limits Y2 and Y3, and dividing by X 2 - Xl' we obtain II, II.

X2 2

It

"r. n(x1, x 2' s,

Xl.!

Y. II.

=_{

t)dsdt

x,

1

x2- X I '

r[lnl!(X+iY2) I-Inl !(X+IYl)ll dX }(YJ-Y2)

x,

x.

+{X2

1

Xl

J

(In 1!(x+iY3) I-In !!(x+IY2)! \dX}(Y2 - YI)

x.

+_1_ f f C(XI' 11, II.

x2- x l

"

II.

x 2 ' s. t)dsdt.

II.

Passing thereupon to the limit as x 2 - Xl - + 00 and recalling that Xl and belong to the relatively dense set E on which the function C(xl , X 2 , s, t) is bounded, we obtain

X2

~(~ (Y2)-1f (YI») (Ya - Y2)+(~(Ya)-~ (Y2») (Y2-YI):> 0

(6.21)

or (6.22)

Thus the function ,rp(y) is convex. By the known properties of convex functions, rp(y) has at each point a right and left derivative, while the right derivative at a given point is not less than the left. It is also known that the right derivative of a convex function is equal to the left at every point with the possible exception of a countable set, and rp'(y) is a nondecreasing function.' THEOREM

4.

If the

derimtil'e q;'(y) exists for Y

= YI

and Y

= Y2

(Y2

> Yl)

then the limit (6.23)

exists. PROOF.

Choose the number h sufficiently small so that

I I

and

+ h2 -

&

~ (Yl -

-~' (Y2)

h) -

rp' (Y1)

I< i I< ; .

In Chapter I we obtained analogous theorems for trigonometrically convex function!.

278

ROOTS OF EXPONENTIAL SUMS

[CHAP. VI

I ntegrate the eq uation (6.20) in s between the limits Yl - h, YI and in t between the limits Y2'}'2 + h, divide by X 2 - Xl' and pass to the limit as X 2 - Xl --. 00, recalling that Xl and X 2 take on values in the relatively dense set E on which the function C(x l , X 2 , s, t) is bounded. We obtain

=

If (Yt) -

'!' (Y2+ h ) - l' (Y2) h

'!' (Yt -

h)

(6.24)

h

obtain

21t

lim

Analogously for h

21t

< 0 we obtain lim

n (XI.

y,. Y2) ?/9 '-. ' ( ) y.) - 9 ' ( YI) -e.

X2' -Xl

X~

~

The desired (6.23) follows from these two inequalities. Moreover, we may free ourselves from the requirement that the quantities Xl and X 2 take on values only in E. Observe that for any Xl and X 2 we may pick out four numbers x~, and belonging to E and such that Ix; - x~1 < I, Ix; - x~1 < 1 and x~ < Xl < X;, X; < x 2 < x~. It follows immediately from the definition of the function n(x l • X 2 , YI,;'2) that

x;, x;, ,2 -

x;

"

x n --. t Xl

X2- X

'"

n (X Yl' , Yl't" -< -----t

(Xl' X 2•

t • X 2•

Y2)

X 2 -X

, , n (XI'

Y2)

X2- X

"

X2•

"

Y,. h)

,

x 2 - xt

"- , XI

x2

and evidently all three quantities have a common limit as x 2 THEOREM

f(x

5.

Iffor Y = Yo

(6.25)

x 2 - xt -

Xl --. 00.

there exists a deriz:ative r'(yo), then the function -q'(yo)'

+ iyo) has a mean motion c,(Yo) =

By Theorem 4 the function n(x l , x 2 ,Yo, t)(x2 - Xl)-l, monotonic in X 2 - Xl --. 00 for all values of the variable t with the possible exclusion of a countable set. Moreover, since that function is bounded for )'1 ~ I ~ Yl + h, there exists a limit PROOF.

t, has a limit as

y,+h

lim ::1:.-.2:,

-+- 00

(X2 -

Xl) -1

f

fl.

n

(Xl' X 2 •

Yo. t) dt •

Sec. 4]

A THEOREM ON MEAN MOTION

Putting s = Yo in formula (6.20) and integrating both sides + h, we obtain

279 In

' . rom

YI to

YI

h arg/(x2+ IYO)-arg!(x,-t- /yo) X2- X !

x,

= - ( X 2 - X 1)-1

J r

Inlflx+i(Yl+h)lldx

X,

X,

+(X2 -X I )-1

In!f(x+(v 1)ldx

'>

X,

y,+11

-(X2 - X1)-1

f

C(XI'

II,

+ 2'1t (X2 -

Xii'

Yo. t)dt

L',+1I

f

Xl) -1

n (Xl' X2• Yo. t) dt.

II.

All the quantities in the right side of the equation have limit~ as "'2 Passing to the limit we obtain from Theorem 4

-

Xl

--+

x:.

(6.27) 4. A Theorem on Mean Motion for an Almost-Periodic Function with Semibounded Spectrum

It follows from (6.27) that etCy) is a monotonically decreasing function. If the greatest lower bound - ~ of the spectrum of the function is finite. then the. function eit!.z fez) is bounded in the upper half-plane, and thus the convex function 9':>(y) - ~y is also bounded for y > O. This is possible only under the condition tp'(y) - ~ ~ O. Thus. in this case the monotonically decreasing function ctCy) is bounded below by the number - ~ and accordingly has a limit lim

cr(Y)~-~'

(6.28)

II~+OO

It follows from Lemma 3 that if the num':ler -~ enters into the spectrum of the function /lx), then equality holds in (6,28). It turns out that this limit is

equal to - ~ also in the case when junction f(x).

~

doe,' not enter into the spectrum of the

This theorem seems very natural, but its proof is rather complicated. s In what follows we shall use the theorem for the study of general exponential sums. For Its proofwe shall need a series oflemmas. First of all we prove the following ge::eral lemma. • B. Ja. Levin (4).

ROOTS OF EXPONENTIAL SUMS

280

[CHAP. VI

LEMMA 6. If the functionf(z), holomorphic in the circle 1= - zol ~ r, satisfies the inequality If(z)1 ~ I and if moreorer I/(zo)I ~ e- T •

(6.29)

then the l'ariation of the function arg}(=) on any rectilinear segment lying in the circle Iz - zol ~ p = r(2e)-1 does not exceed the quantity (4 + 217)Y. PROOF. Putting rr = re-I and denoting by nU, cr, zo) the number of roots of the function fez) in the circle 1= - zol < cr. and using Lemma 4 of § 5 of Chapter I, we obtain (6.30) n (/. o. zo)

< "(.

If we set up for these roots the function

X(z, zo) where u

=z-

Zo

and u"

= air <£I (z.

=

II

I uk I < a

a(u -

Uk)

a2 _

Uk U '

Zo (u" a root off(z», .he function

zo) = fez) x- 1 (z. !o).

(6.31)

will be holomorphic and will not have roots in the circle Iz - zol <; cr. and will satisfy on the boundary and therefore inside this circle the inequality

19 (z. zo) 1< 1 Since inside this circle Ix(z, =0)1 < 1. therefore Iq{zo' zo)1 > e-

Y•

Applying to the function In q{z. zo)/,,(zo. zo). holomorphic for Iz - zol the Caratheodory inequality (Chapter I, § 6) with R = 2r. we obtain

< cr.

/In'f(z. zo)-In'f(zo. zo)I<-2Inl<£l(zo. zo)1 for I Z -

Zo 1< .;- = p.

Thus. if Zl and Z2 are two points of the circle Iz - zol

< p. then

lin <£I (Z2' zo) -In 9 (Z1' zo) 1 < 41 and moreover

<

I arg 9 (Z2' zo) - arg 'f (Z1' zo) 1 41· The variation of the argument of the function cr(u - u,,)/crl rectilinear segment is smaller than 217. so that largx(z2' zo)-argx(zt. zo)I<2M(/. o. zo)'

(6.32) -

u"u on any (6.33)

Comparing (6.31), (6.32). (6.33) and (6.30), we find that 1arg/(z,J -

arg!(zl) 1 < (4+ 21t) 1·

Here it evidently makes no difference whether one takes as argf(z) the right or left argument as the point z moves over the segment [ZI' zJ. If the spectrum is

Sec. 4]

281

A THEOREM ON MEAN MOTION

bounded below, then one may assume without loss of generality that its lower bound is equal to zero. If in addition zero does not enter into the spectrum, then from Lemma 3 the function I(x + iy) converges uniformly to zero as y - + 00. Therefore without loss of generality one may assume that /I(x)/ <; 1 and thus that in the entire half-plane y ;> 0 we haye If(z)/ <; 1. To estimate the variation of the argument of the function I(z) on the line y = R we use Lemma 6. In order to do this we must prove that in the neighborhood of each such line we can find a sufficiently dense set of points (supporting points) in which an inequality of type (6.29) is satisfied. The following material is devoted to the proof of the existence of such supporting points. LEMMA

7.

Suppose that {at} is a sequence o/numbers CD

X(Z)

I

~ 11m ~" < 00

1m a" > 0,

"=1

and

= ir(l--=-)(l _ ~)-1 "=1 a" a"

(6.34)

.

Then the quantity

J (X, R) =

r.2J?

f

In I X(Rei&) Isin fj d6

o

cont'erges to zero as R increases. PROOF.

Noting that /X(x)/ = I and using the Carleman formula (5.02'), we

may write

~ J (X, where at

= r~iO••

R)

+i 1m X' (0) = ~ (r~ -

But X'(O)

=

~) sin 611 ,

r" <.IJ "" -2; I 1m l/al: and accordingly 1:=1

I

'2 J (X, R) =

~

sin 6"

~

r;;- -

r,,>.IJ For some r. > U v'e wHI have

,!.

~-

~.

1

R'~ ~

rIc SIn

r,,<.IJ

~

~

~ r"sin6" < l.J r.r.

Moreover, for some R.

so that asymptotically

6"

>-

sin II" rIc

1 ~

~

l(A

.

rIc SIn

6".

r,,
< ~2·

> 0 and R > R" ~ ~ r"sinfJ,,<-i, r,,
- 2e < J("1... R) < O. The right inequality results from the inequality lX(z)1

< 1 for 1m z > o.

282

ROOTS OF EXPONENTIAL SUMS

[CHAP. VI

LEMMA 8. If the function 1f'(z) is holomorphicfor 1m Z > 0, is continuous in the closed half-plane, is of exponential type, and satisfies ,he inequality 00

(6.35) then

lim J('f, R) = k. R-+- 00

PROOF. Setting up the expression J(1f', R) and changing the order of integration, we obtain 00

J('~.

R)=;

f

Inly(t)lk(t. R)dt+k,

-00

where

f"

1 sIn28 dO k(t. R)=-:;:. R2+t2_2RtcosO' o

which 'on calculation yields

>(/.

R)~\

l

Thus,

for

It I < R.

for

It I> R.

212

R

J(Y.

R)=Tt~a

f

Inl'Ht)y(-t)ldt

o

f

00

+ Tt

-.!..

In 1'I-(t)",(-t) 1 dt+k.

(6.36)

t2

R

The existence of the second integral may be obtained immediately from the existence of the integral (6.35) by integrating by parts,just as in § 3 of Chapter V. Introducing the function

,.

'fl (x) =

J. In I 'I- (t) (~H -

t) I dt

(X> A).

).

we easily find that R

f

_1_ R2 ).

R

In I y(t) Y(- i) Idt = 'fl (t)

R

:a 1- ~a S

'fl (t) t dt,

l

l

Sec. 4]

283

A THEOREM ON MEAN MOTION

and recalling that Ifit) tends to zero for t some A. > 0 and R > ). ~ A.

> A and

B

J

11t~2

In 1y(t) Y( - t) 1dt

A ->-

00,

we find that for

I< ; .

(6.37)

l

Choosing the number R. sufficientlv large, we will also have for R l

11t~1I

f

In 11\1 (I) Y( - t) Idt

o

> R,

I< i (6.38)

and co

SIn 1 (I)t a1jI( 11t' 1

<\I

I) 1dt I

< ~3·

B

The assertion of the lemma follows from (6.36), (6.37) and (6.38). Combining Lemmas 7 and 8 we get LEMMA 9. If the function f(z) is holomorphic and of exponential type in the half-plane 1m z ~ 0, while 00

dt S Inlf(t)1 1+ t

(6.39)

lim In If(ly) 1= k, + co y

(6.40)

lim J 1../, R) = k.

(6.41)

2

-co

exists and

y~

then

B~co

PROOF. From Theorem 5 of § 3, Chapter V, it follows from the hypotheses of this lemma that the functionf(z) may be represented in the form

/(z) = y(z)X (z), where x(z) is defined by formula (b.34) (the ale are the roots of the function /(z» and tp(z) admits a representation (6.35). Therefore we obtain

J(/, R)=J('f, R)+J(X, R),

(6.42)

and the result follows from Lemmas 6 and 7. LEMMA 10. Suppose that f(x) ;s an almost-periodic function with 'he greatest lower bound of its spectrum equal to zero, and let

•

J(f,

'1:,

R)

= :R

f

o

In 1/('1:+ Reft) 1sin 8 d8.

(6.43)

284

[CHAP. VI

ROOTS OF EXPONENTIAL SUMS

Then to each number £ > 0 there correspond a relatit'ely dense set of real numbers and a number R. such that for R > R.

T

-I

< J(f. 'to

R)

<0

(6.44)

for the entire set of numbers {T}. PROOF. Without loss of generality one may suppose that If(x) I < I (it is sufficient to multiply the function f<x) by some constant). From CarIeman's formula we have

B

~

f Inlf(t+~)f(~-t)l(t~

-

~~)dt

A

= J(f. 'to R)+ Adf. 'to R).

A,(f. <.

R)~ -1m !

j

10/«+ ,e~)J( ~:

(6.45)

- e~.) ...

The left side of equation (6.45) is a function monotonically increasing as T. We will denote it by 'P(R, T). From Lemma 9, we have for some R; > 0 and R > R;

R increases for any fixed

-~

< JU. o.

R)

< O.

(6.46)

Suppose moreover that R; is so large that for R ~ R; the inequality

I BA (f. o.

R) I

<~

(6.47)

is satisfied, where

Bdf. ~. R)=AA(f. ~. R)-Adf. ~. 00)

= -1m 7t~

f"

(Inf('t+

Aei8)JAe~8dO.

o

Choose the numbers 0 < A < 1 and R. ~ R; > A so that on the semicircles and z = R.eill (0 1T) there are no roots of the function f(z). Denote the greatest lower bound of If(z)1 on these semicircles by m > 0 and choose the number () so that lJ < m./12. If T is a lJ-shift of the functionf(z) in the strip 0 y R and the point z belongs to the circumference of one of the semicircles, then f(Z)-/(Z+~)JI 26 £ (6.48) II nf(z+'t)- lnf(z)1 = In 1I(z) 6'

z

= kill

< () <

< <

I[

<m<

Sec. 4]

285

A THEOREM ON MEAN MOTION

Thus for R

~

R, and the indicated choice of the numbers

IJ(/,

RJ-J(/, 0, RJI

't,

'T

<~ } (6.49)

and

I B).(/,

't,

RJ 1< ~ .

R.)-B).(/, 0,

From the first of these inequalities and (6.46) it follows that

2£

-""3 < J(/,

't,

R.)

< O.

(6.50)

From Carleman's formula the function J(f, 'T, R) is the sum of a monotonic increasing function "P(R, or) and a function A,.
-8<J(/,

't,

R)
(6.51)

where {or} is a relatively dense set of l5-shifts of the function fez) in the strip '" R. The lemma is proved. From Lemma IO it easily follows that on each arc z = Rei6 + 'T, 10 - {-17/2)1 < ex (ex > 0) and R > R. there exists a point ZT at which

o '" y

(

1)

=

ttl. )

(6.52)

4 Sin CI •

Indeed, in the contrary case we would have

" 11 2+

J(/,

't,

R)

f

< ttR2

In II (Re i9 ) I sin 6 d6

"

--II

2

< -.!1) sin ex = tt

-8,

(6.53)

which contradicts (6.44). Thus from the last four lemmas we obtain a corollary which we formulate as a separate lemma. LEMMA II. If the greatest lower bound of the spectrum of the almost-periodic function f(x) is equal to zero, then to each pair of numbers E > 0 and { > 0 there correspond a relatively dense set of numbers {or} and a number R(E, {) > 0 such that if R > R(E, {) each circle with center at the point 'T + iR and o/radius {R contains some point z. at which

(6.54) The preceding four lemmas were necessary only in the proof of Lemma II, of which we now take advantage. Using Lemmas 6 and II we may prove the following theorem.

ROOTS OF EXPONENTIAL SUMS

286

[CHAP. VI

THEOREM 6. If the greatest lower bound of the spectrum of the almostperiodic function f(x) is equal to - tl, then 6 lim cf(Y) = .l. (6.55) y~+c.o

PROOF. One evidently has to consider only the case when the number - a does not enter into the spectrum. Without loss of generality we may take tl = 0 and l/(x)1 ~ I (- 00 < x < 00), since if necessary the functionf(x) may be multiplied by kei.l x , with k a constant. Suppose that E > 0 is given. Choose a number ~ > 0 suclt that t < (1 - O/4e, define a relatively dense set {'T} and a number R(I!, 0 as in Lemma II, and choose a number R: ;;;. max{ R(I!, n, 4e/[V3 (1 where / is the enclosing interval for the set 'T. We describe around each of the points ZT' whose existence was asserted in Lemma II, a circle K.,R of radius d = R(2e) -1(1 - 0 with center at that point. With the indicated choice of the number t and R > R; these circles completely cover the line y = R. From Lemma 11 we have If(z,)1 > e- 0 andy> R; (6.56)

on

and accordingly (6.57)

lim c,(y)=O. y~+oo

From the theorem just proved and from Theorems 4 and 5 we obtain the

If the spectrum of an almost-periodic (unction

COROLLARY.

is bounded and

then lim a (Yl' Y2) =

~

(6.58)

,

I"

~-oo Y.~+oo

where d is the length of the smallest segment contammg tne entire spectrum of the function f(x). • Here, as usual, c,(y) denotes the mean motion for the function/(x Cf

(Y)

=

· 11m

oX,-x,-"+co

argl (X2

+ Iy) X2 -

arg l(x I Xl

+ iy), i.e.,

+ Iy)

Sec. 5]

FUNCTIONS APPROXIMATED BY EXPONENTIAL POLYNOMIALS

287

In fact, by Theorem 4 U(Yl' Y2) exists for all values Yl and Y2 except possibly a countable set, while U(Yl, Y2) = 1p'(Y2) - 1p'(Yl)' or what is the same thing, 1 a(Yl' Y])= 2'11: (C,- - 00 or Y2 -+ + 00 the quantities C,(Yl} and c,(Y2) tend, respectively, to the lower and upper bounds of the spectrum of the function.' S. Functions Approximated by Exponential Polynomials The theory presented in the preceding chapters makes it possible to carry out a very thorough investigation of the question of the distribution of the roots for entire functions of exponential type of the form (6.01) for which the exponents AI; belong to the boundary of somr. bounded convex region 1. 8 In this investigation we make use of the results presented in the first four sections of this chapter. We shall carry out an investigation for functions of a wider class than those represented by the series (6.01). In order to define this class we introduce a norm for entire functions of exceptional type which is connected with a convex region I. Let [be any bounded convex region and h( - 0) its supporting function. By means of h(O) we define a norm 11£111 of the entire function F(z) of exponential type in the following way:

IIFIII=

sup

,.>0,

(IF(re")1 e- 1d8 ) r).

(6.60)

0<8<21<

The set of entire functions of exponential type which are bounded in this norm forms a linear space BI . If a sequence of entire functions bounded in this norm converges in the norm, then it evidently converges uniformly in any finite portion of the. plane, and accordingly there exists a limit entire functioa with bounded norm. Thus to each bounded convex region [ there corresponds a complete space B I of functions of exponential type. In it are mcluded only those entire functions of eXPQnential type with indicator diagrams which belong to the convex region I. (I is symmetric to the region [relative to the real axis.) It is easy to see that all exponential polynomials (6.00), whose exponents belong to the boundary or 7 One may show that if one of the ends does not enter the spectrum, then a(y .. Y.) < !.... for any y .. y. (y. > Yl)' 2,.. • This question has been investigated in many papers. See, for example, M. Rcgensburger [IJ and O. Mattison [IJ.

288

[CHAP. VI

ROOTS OF EXPONENTIAL SUMS

the region 1, enter into BI and IIPIII <; l:~ the relations

latl. This follows immediately from

n

n

1P(reit) I <: (~ 1ak I) max ell" I reo. (8+_,,' = (~l ak I) ell (8)r 1

k

1

(ak:;:= arg Ak)'

In exactly the same way one shows that a function f(x) of type (6.0 I) belongs to BI if as the norming region lone takes the smallest convex region containing all the exponents Alt. It is also evident that the series (6.01) converges in the norm of BI • We define the class EI of entire functions approximated in the norm of BI by exponential polynomials whose exponents belong to the frontier of 1. This class of functions is wider than the class of functions represented by the series (6.01 ). We shall now find a representation of functions of the class EI which is important for what follows. Let A be any point of the boundary of l, and fJ a direction such that the point Ais a supporting point for the line 1,: x cos fJ + y sin fJ - h(fJ) = O. The direction arg z = fJ perpendicular to the line I, will be called the direction corresponding to the point A. If A is not an angular point of the boundary of the region i, then the direction corresponding to it is unique. In the opposite case directions form an angle corresponding to the point A. Construct the function F~ (z) = F (ze i8) exp (- Aei8z),

(6.61)

where A is a point of the boundary of 1, and (J is a direction corresponding to this point. The conjugate diagram of this function is obtained from the conjugat/'" diagram 1 of the function F(z) by shifting by the vector -A and rotating through the angle O. Here the point A goes into the point zero, the entire diagram is situated in the left half-plane, and the imaginary axis is supported by at the point zero. The supporting function of the region I; will be denoted by h:(cp). In the general case we will have a supporting segment belonging to the imaginary axis and containing the point zero. Each exponential polynomial P! .•(z) may be represented in the form

I:

1; 1;

P..• (z) = ct., •(z)+ R~, • (z)

(6.62)

in such a way that the exponents of the exponential polynomial Q! .•(z) lie on a segment of the imaginary axis, and the real parts of the exponents of the exponential polynomial R!.•(z) are negative. For the given situation of the diagnam ""'(0) == 0, so that since the sequence of polynomials P..(z) converges in norm, we have for x > 0 and n, m > N.

I:

1P~, • (x) -

p.., •(x) I < •.

Sec. 5]

FUNCTIONS APPROXIMATED BY EXPONENTIAL POLYNOMIALS

On the other hand, R~.o(x) -+ 0 and R!.o(x) -~ 0 for x -+ that (n, m lim I Q);/I. 8 (x)- Q~.8(X) I ~ £

289

+ 00, which means

> NJ.

:Il-'+<XI

Hence for the almost-periodic polynomials Q~I.O(X) and Q!.o<x) we have the following inequality, valid on the entire. real axis:

IQ~. a(x)- Q~.8(X) 1< £

( - 00

< x < 00).

Thus the sequence of almost-periodic polynomials Q!.o(z) converges uniformly to an almost-periodic function tp~x) and we have obtained the following lemma. LEMMA 12. Let F(z) be an entire function to which a sequence of exponential polynomials P,,(z) converges in the norm of B]> the polynomials P,,(z) having exponents belonging to the boundary of I. Let F:(z) be the function defined by equation (6.61). Then F:(z) may be represented in the form

~OO=~OO+~~

~~

where tp~(z) is an almost-periodic function whose spectrum lies on a portion of the boundary of the region lying on the imaginary axis, and ~(z) converges uniformly to zero as Re z -+ + 00 and 11m zl M (M any positive number).

I:,

<

In particular, it may turn out that 'I'~(z)

= o.

REMARK 1. The representation (6.63) is unique. Indeed, from the equality of the two representations

4>1~ (z)

+ 'tl~ (z) =

«P2e (z) + '1~ (z)

it follows that

lim (·!,~o (x) -. ,~~ (x») = 0 I T '

x ...... ao

and in view of the almost-periodicity of the functions tlto(x) and tl2II(x) we obtain l

'11& (x)

= ~l (x).

REMARK 2. If A is an extreme point of the region 1, then the function tI.
while lim T _ co lA[(r + iT)e i '1 = 0 and the convergence is uniform in Tfor ITI <; M, ~ (J is the direction corresponding to the point A of the boundary of the region 1, and such that the corresponding supporting line has only the one point A of contact. Now we shall define the important concept of the spectrum of a function of class E 1 •

290

ROOTS OF EXPONENTIAL SUMS

[CHAP. VI

DEFINITION. The point A of the boundary of the bounded region J i~ said to be a point of the spectrum of the function F(z) of class E[ if zero is a point of the spectrum of the almost-periodic function 'f'~ in the representation

F (zei~) exp (- Azei~)

in which x--+

(J

= '1~ (z) + IP~ (z)

is the direction corresponding to the point A with ~(x) -- 0 when

+00.

The collection of all points of the spectrum of the function F(z) is called its spectrum. From the uniqueness of the representation (6.63) it follows that the spectrum of the function F(z) of class E[ does not depend on the choice of the approximating sequence of exponential polynomials. We shall show that if A is a point of the spectrum of the function F(z), then for some NA > 0 and n > NA all the exponential polynomials P,,(z) contain ;. among their exponents and

lim an (A)

=

(6.65)

a (A),

where a,,(A) is the coefficient of eAz in the exponential polynomial P,,(z), and a(A) is the free term in the Founer series of the almost-periodic function 'f'~(z). Indeed, if A is an extrcme point of J, then from (6.64) and the inequality

iF (re iij ) -

P1l (re i9 ) I e- h (9) r

< a,

which is equivalent to the inequ1 1;ty

I exp (- ).rei~) IF (re i6 ) - P n (re iQ ») I < a, one obtains on passing to the limit as r --+

00

Therefore (6.65) follows immediately. If now A is a point of a rectilinear segment of the boundary of J, then (6.65) follows from the representation (6.63) of the function Ft(z) and the known property (f) of almost-periodic functions. Conversely, if (6.65) is satisfied, then the point ). enters into the spectrum of the function F(z). Thus we obtain a new definition of the spectrum. The spectrum of the entire function F(i), approximated in the norm of B] by exponential polynomials whose exponents belong to the boundary of I, is the set of those points A of the boundary of 1for which the quantity a (A)

=

lim an (A) n~oo

is not equal to zero. Thus the spectrum of the function F(z) does not depend on the choice of the norming region I, whose boundary contains all the exponents {AIo} of the

Sec. 5]

FUNCTIONS APPROXIMATED BY EXPONENTIAL POLYNOMIALS

291

approximating sequence of exponential polynomials. In particular, the spectrum of the entire functionf(z), represented by the series IX)

fez)

=

~ a,l'"~

(6.01)

"~1

in which the numbers At belong to the boundary of a convex region, consists of the set of all exponents At. From the representation (6.63) it is easy to obtain an expression for the "Fourier coefficient" of the function F(z), and indeed to obtain the formula a (A) =

lim

7'4+~

~

7'

f

F (re n ) exp (- Aei8r) dr

0

(6.66) '

where A and 0 have the same meaning as in (6.61). Thus the spectrum of the function F(z) may also be defined as the set of values A for which a(A), defined by formula (6.66), is distinct from zero. From the representation (6.63) it also follows that if Ais a point of the spectrum of the function F(z), then the point 1 enters into the boundary of the indicator diagram of the function F(z), and accordingly the conjugate diagram of the function F(z) contains all the points of its spectrum. Observe that the conjugate diagram of a function F(z) of cla,ss E[ does not necessarily coincide with the smallest convex region containing the whole spectrum of the function. To see this, we construct an entire function of the class E[ which has no spectrum. Let fez) be a function holomorphic outside the unit circle, not continuable through the unit circumference, equal to zero at infinity and continuous for Izl ;> 1. 9 Put

F (z) =

J

eUf(A) dA,

where C is the unit circumference. Evidently F(z) is an entire function of exponential type and its conjugate diagram is the unit circle. Put fa

Pn (z) = ~

ile' f(A,Ie)(AIe+1- Ale)

Ie-I

We shall show that if IAki-l - Atl < 6 (k = 1,2, ... , n) and lJ -+ 0 the function p.(z) tends to F(z) in the norm of Eb if as I we choose the unit circle C We may without loss of generality as§ume that z = r, and consider the quantity

IF

(r)- p. (r)

I'-·I~.~ [1"'-""(')-"'-""(''>1 d'i'

• For example one may take ex>

f

(z)

=~. II-I

'z,. ..

n9

292

[CHAP. VI

ROOTS OF EXPONENTIAL SUMS

where s" is an arc of the circumference of C between the points A" and A"+l (k = 1,2, ... ,n). We have fa

IF (r)- P n (r)1 e- r ~ ~

f Ie\~-I)r II (i..)- I (i.,,») II I I)rll l(i·,,) i I d)' I+~ f II (i.,,) II d)·I. d).

kml

't

+ 1: f 1/-" 'Ie,

11 ,- -

e(l_" ..

"

•

"

Ilk

s; s;

where s~ is the arc of the circumference on which larg AI ;) TJ > 0, and is the remainder of the arc. We choose the arcs sit so small that the oscillation of the functionf(A)'on each of these arcs will be less than E/677 and so that if A E and r > 0 we will have I e(l-~)J- - e(A.-I), I 6:'M '

<

where M = max I[(A)I for A E C. Choosing in addition TJ

IF(r)-P,,(r) Ie- r < i

< E/677. we obtain

and in view of the equivalence of all directions

I F (re ie ) -

P" (rei.) I e - r

<

!.

Thus F(z) is of class E[. On the other hand. it is easy to show that for any value of () r-+oo

i.e., the spectrum of the function F(z) is empty. In connection with this example the following theorem takes on particular interest. THEOREM 7. If the norming region [ is a polygon, then a function F(z) of class E[ may be represented in the form

(6.67)

where ~ is a point on the side ~ of the polygon i, the angle CPj defines the direction of the normal to that side, and 'h(z) is an almost-periodic function, while the whole sped",!!" of the function 'h(z) lies on the stK!'ent into which the side ~ of the polygon I goes on displacing it by the vector - ~ and rotating it by the angle - 'l'i. In addition the set of singularities of the associated junction f(z) coincides with the closure of the spectrum of the function F(z). PROOF. The exponential polynomials P ,,(z) approximating the function F(z) may be represented in the form

PC) n Z =

p ~ ~

1-1

(i._) e Lz, J Yill ze -J •

Sec. 5]

FUNCTIONS APPROXIMATED BY EXPONENTIAL POLYNOMIALS

293

where )'; and 'P; have the same meaning as in the formula (6.67) and tp;n(z) are almost-periodic polynomials, while the entire spectrum of the function tpi"(z) lies on the segment into which the side d; of the polygon I goes under displacement by the vector and rotation by the angle -9/;. Since the sequence P n(z) conv:erges in the norm of B[> therefore for any f: > 0, some N. and n, m > N,

-.xi

(0 .,:;:

fj

< 27":, r > 0).

Hence we immediately obtain lim !'?jn (r) - 'Ijm (r) I

r-+oo

and accordingly

r~oo

~jm (x) I < a

I'}}I. (x) -

(-oo<x
On passing to the limit we obtain (6.67). In order to prove the second part of the theorem it is sufficient to show that the set of singularities of the function associated to the almost-periodic function tp(z) with a bounded spectrum coincides with the closure of the set of its Fourier exponents {iA k }. Denote by X(z) the function associated to the function tp(z) and by .,. Xn (z) = ~ an (A/c) "~lZ-0"k

the function associated to the almost-periodic polynomial tp,,(z) of a sequence uniformly converging to tp(z). From the inequality "t(x)-~n(x)1

it follows that

<

It

(-00

< X < 00)

< ':1

IX(z)-Zn(z)1 On the other hand, lim xXn (x :1)-+0

+ LA,,) =

an (A k ),

and accordingly if ).1; is a point of the spectrum of the function tp{z), then lim :1)-+0

xz (x + LA,,) =

a (A,,)+ O.

Thus, if ).1; is a point of the spectrum of tp(z), then i).1; is a singular point of the function X(z). Suppose conversely that the point 1)'0 lies at a positive distance d from the set {i).I;}. We shall show that iyo is a regular point of the function x(z). Indeed, from the inequality

I ~n (x) it follows that

~m (x) r< a

(-oo<x
11... (z)-1..(z) 1

< 1:1'

294

ROOTS OF EXPONENTIAL SUMS

[CHAP. VI

and applying the Poisson formula to the function In IXn(z) - Xm(Z) I in a circle with center at the point iyo and of radius R < d we obtain

f

21C

1 In I X"(z)-X,,,(z) 1< 21t

R3-pl

a

In RI cos 4'1 R3+ pll-2Rp cos (Ij._ O)dl\l.

o

(z =

1,v9+ pet'),

From this inequality follows the estimate

In Ix"(z)-y',,,(z) I < (In

N+ ~ +: In 2).

from which one immediately finds that the sequence {Xn(z)} converges uniformly in the neighborhood of the point iyo, so that there is no singularity of the function X(z) at the point iyo. The theorem is proved. Another example of an entire function for which the closure of the spectrum coincides with the set of singularities of the associated function, so that its conjugate diagram coincides with the smallest convex region containing the spectrum, is given by a function represented by the series <.0

fez)

= k=l ~ aki1f6

(6.01)

in which l::'=l lOti < 00 and the At belong to the boundary of the convex region I (Chapter I, § 21). 6. Growth of a Function of Class E 1 for a Norming Region in the Form of a Polygon In this section we consider the case mentioned at the end of the preceding section. Namely, we suppose that a sequence of exponential polynomials converges in the norm (6.60), while the norming region I is a polygon and the exponents of the exponential polynomials approximating the function F(z) lie on the boundary of that polygon. Our fundamental result consists in the fact that a function F(z) defined in this way is a function of completely regular growth. First we prove the following lemma. LEMMA 13. Suppose that 971(Z) and 97,,(Z) are entire almost-periodic functiOns of exponential type with non-negative spectra. Then the entire function of exponential type F (z) =
is a function of completely regular growth. PROOF. Without loss of generality we may assume that the point z == 0 does not enter simultaneously in the spectra of both functions 971(X) and 97.(X).

Sec. 6]

GROWTH OF A FUNCTION OF CLASS

E[

295

Since the functions associated to the functions IPI(zr''X') and 'f2{zr icx2 ' have no common singularities other than perhaps the origin, the conjugate diagram of the function F(z) is the smallest convex region containing two rectilinear segments d. and d2 on the rays arg z = 'IT /2 - a. and arg z = 'IT /2 - a 2• These rays are obtained by rotating the smallest segment of the imaginary axis containing the spectra of the functions CP.(z) and CP2(Z), by the angles a. and a 2• The functions IPI(Z) and 1P2(Z) are bounded on the real axis and therefore (Chapter V, § 4. Theorem II) are functions of completely regular growth. Thus the following limits exist: i(ij-mi) I lim" lnj1';(re =h(&) (j = 1. 2). r J Evidently if for some value of the angle 8 we have h1(8) =F h2(f), then the limit

exists. If the point z = 0 is not a common end of the segments d, and d2 , then there exist only two values of the angle I} for which h1«() = h2«(). These values of 8 correspond to the directions of the normals to those sides of the indicator diagram of the function F(z) which join the ends of the segments dl and d2 • The function F(z) is of completely regular growth in all directions except possibly two, and. by the theorem that the set of directions of completely regular growth is closed, it is a function of completely regular growth in the entire plane (Chapter III. corollary to Theorem 1). If the point z = 0 is a common end of the segments d1 and ~, then h1«() = h,.(0) for al " 0 " 'IT + a2. We must prove that F(z) is of completely regular growth inside this angle. Suppose that the point z = 0 is a point of the spectrum of the function F(z) with Fourier coefficient ao. From the representation (6.64) of this function it follows that for a. < 0 < 'IT + a2 lim F (,e il ) =

ao·

Hence it follows that the function F(z) is oT completely regular growth inside that angle, and thus in the entire plane. The case when the point z = 0 is a common end point of the two segments dl and d2 and does not enter into the spectrum of the function F(z) is somewhat more complicated. Suppose, without loss of generality, that (Xl = 0, so that F (z) = 1?1 (z)

+

qljl (ze- ill.).

(6.69)

It is easy to see, using the approximation of the function 9'1(Z) by generalized

296

[CHAP. VI

ROOTS OF EXPONENTIAL SUMS

trigonometric polynomials, that as x ->- + 00 P2(Ze i>,) converges uniformly to zero. From Lemma I of this chapter, in any horizontal strip Iyl <; h, but outside circles of radius c5 with centers at the roots of the function /f'1(Z), the inequality

>

I '-?1 (z) I-'"~· m (~, h) 0 is satisfied. Since for some Xo > X(~, h), x> Xo and Iyl ~ h the inequality Ir2(u- i "')1 < m(c5, h) holds, therefore in the strip Iyl ~ h for x > Xo there are no roots of the function F(z) outside the regions formed by the exceptional circles. By Lemma 2, the number of roots of the function P1(Z) in the rectangle Iyl ~ h. I ~ x < t + I (t arbitrary) does not exceed some number kll • For c5 < [2(k/l + 1)]-1 the diameters of the regions formed by the "clusters" of exceptional circles do not exceed the quantity 2kh~' so that for a sufficiently small d they will be arbitrarily small. By the well-known Rouche theorem the function F(z) has, in each such region, the same number of roots as the function fP1(Z). Denote by nF(h, t) the number of roots of the :"unction F(z) in the rectangle o < x < t, Iyl ~ h, and by n(~(u, t) the number of roots of the function fPl(Z) in the rectangle Iyl .;;; u, 0 < x ~ t. From the above we have for TJ > 0 the following inequality: lim

n'l) If

(h -." t) t ""

t -+00

-< tlim

n~)(h

+

1),

t)

t

(6.70)

-+.x>

The limits referred to in the inequality (6.70), which exist in view of Theorem 4 of this chapter, are continuous functions of h + TJ and h - TJ for all values of these quantities with the possible exclusion of a countable set. Thus, for almost all values of the quantity h there exists the limit .

n F (h, t)

t-.co

t

11m

=

\.

1m

n~1J (h, t)

t

t-+oo

Making use of the corollary to Theorem 6 of this chapter, we may assert that for any E > 0 and h > h. nF(h. t) It -. . I1m -(6.71) t -++00 t 2x •

>

where 11 is the length of the spectrum of the function fP1(Z). Thus, the lower density of the set of roots inside the angle (-E, E) (E containing the positive half-axis, satisfies the inequality

r

1m

r~

nF(r, -1,1)

r

I

~_l.

2K

> 0),

Sec. 6]

GROWTH OF A FUNCTION OF CLASS E1

297

Analogously, for an angle containing the normal to the second side of the indicator diagram. .

nF(r, Tt-Cl2- I , Tt-Cl3+a)

hm

r

.... co

'.

~-2 ' 11:

(6.72)

where 12 is the length of the spectrum f1'2(Z). Outside the angle (0, rr - ~) the function F(z) is of comph:tely regular growth. Therefore inside a small angle, containing the normal to the third side of the indicator diagram, there exists a density for the set of roots of the (unction F(z) and it is equal to the length of the third side 13 , divided by 2rr (Chapter III). Thus, . nF(r) ....... sF I1m (6.73) --~r "211;' "~oo

where SF is the length of the entire indicator diagram of the function F(z). By Theorem 3 of Chapter IV and the remark to it, F(z) is a function of completely regular growth. REMARK. In (6.71) we have also established the following fact relative to the asymptotic distribution of the roots of the function F(z). In every strip parallel to the normals d;, 11m (ze-iOl;)1 <; h, there exists a density for the set of roots of the function F:(z) for all values of h with the possible exception of a countable set, while for an unbounded expansion of the strip (h - + 00) this density converges to the quantity Ij /2rr, where I; is the length of the corresponding segment. THEOREM 8. If I is a polygon, then the entire function F(z) of class E1 is a function of completely regular growth. PROOF. Consider the angle between the normals arg z =(Jj and arg z = 0i+l to the two successive sides dj and d;+l of the indicator diagram IF of the function F(z). Without loss of generality one may assume that the common end point of the segments d j and di+l lies at the point z = O. The function F(z) may be represented in the form

P (z) = IJIJ (ze-"j) +lJIj+1 (ze-i~j+l) + PM+! (z),

(6.74)

where tpj(z) and tpi+l(Z) are almost-periodic functions with positive spectra (one of them may be equal to zero), and FiJ+l(z) is the sum of the remaining terms of (6.61). The first two terms form, by Lemma 13, a function of completely regular growth with an indicator equal to zero inside the angle (Jj < arg z < OJ+l' The conjugate diagram of the function F;.j+l(z) is a polygon lying in the smallest convex polygon containing all the sides of the polygon I other than dj and dj .Ll' Therefore the indicator of F;.i+l(Z) is a negative quantity inside the angle OJ < arg z < (Jl+l' a~d accordingly F(z) is of completely regular growth inside that angle. Thus F(z) is of completely regular growth inside any angle between

298

ROOTS OF EXPONENTIAL SUMS

[CHAP. VI

two successive normals. Since the set of rays of completely regular growth is closed (Chapter Ill, Theorem I), F(z) is of completely regular growth in the whole plane. The theorem is proved. Comparing this theorem with Theorem 2 of Chapter III, we find that all the roots of the function F(z) except possibly for a set of zero density lie inside arbitrarily small angles containinf.: the normals to the sides of the indicator diagram, and the density of the set of roots inside each of these angles is equal to 1;/27T, where I; is the length of the corresponding side. In a special case Theorem 8 may be substantially sharpened. This sharpening may be formulated as a separate theorem. THEOREM 9. If F(z) is the entire function defined in Theorem 8, and all the vertices of the polygon 1 enter into its spectrum, then all of its roots lie in the half-planes 11m (ze-iO;)1 < h, Re (ze- iO;) ~ 0, where 0; defines the direction of the norma/to the side of the polygon I and h is some positive number. Moreover, if one denotes by n;(t) the number of roots of the function F(z) in the rectangle '8

i 1m (ze-' i) 1<.. h u 0 <.. Re (ze

i9

i)

-<. t.thenni(t) =

dJ

211: t + 0(1).

PROOF. This theorem may be proved independently of the general Theorem 8. Withoul loss of geilt:raiity one may assume that the vertex Ai' in which the sides d; and d:: 1 of the polygon come together. lies at the poi!:t O. From the representation

F(z)=aO+~i(ze-i9j)+l\Ii+1(ze i9i + 1 )+Fi . j +1(z)

(6.75)

(in which 'I';(z), 'I';+1(z) and F; ;+1(z) have the same meaning as in (6.74», it follows that for some h > 0 th~re are no roots of the function F(z) inside the angle Im(ze-~) ;> h, Im(ze-~+') ;> h (~+I > ~). Indeed, the spectrum of the almost-periodic function 'I';(z) is positive, and accordingly the function uniformly converges to zero as 1m z ---+ + 00. Thus. the second term in (6.75) tends to zero as 1m (eiO;z) ---+ + 00. Analogously one proves that the third term in (6.75) converges uniformly to zero as 1m (ei6/+ l z)---+ + 00. In exactly the same way one easily proves that F;.i+1(Z) ---+ 0 uniformly for 1m (eill1z) :> h, 1m (~i61+IZ) :> h, and h -- + 00. Thus, in the indicated region for sufficiently small h we have I F(z)-aol I aol. (6.76)

<

and thus F(z) has no roots. In order to obtain an asymptotic estimate of the number of roots we shall suppose without los,; of generality that F (z) = '~i (z) FI (z).

+

where 'PAz) is an almost-periodic function of class [&], and F1(z) -- 0 uniformly as Re z -- + 00 in any strip 11m zl <; h. From Theorem 3 of this chapter, all the

Sec. 6]

GROWTH OF A FUNCTION OF CLASS E[

299

roots of the function 'P;(z) lie in some strip 11m zl ~ hand n;(t) = (/;/27T)t + 0(1), where I; is the "length" of the spectrum 'P;(z). and n;(t) is the number of its roots in the rectangle 0 < Re z ~ t, 11m zl <; h. Moreover, outside regions covered by circles of radius () with centers at the roots of the function 'P;(z), it \s true that I'P;(z) I ;> m(l5) > O. For some Xo = x«(). Re z > Xo and 11m zl < h we obtain IF}(z)1 < m(b), and accordingly the function F(z) has no roots outside these regions. By Rouche's theorem the function F(z) has in each of these regions the same number of roots as the function 'P;(z). Thus, (6.77)

REMARK. Using Rouche's theorem it is easy to prove that the asymptotic picture of the distribution of the roots of the function

F(z) =

F (z)

+ (z),

(6.78)

where F(z) satisfies the conditions of Theorem 9, and (z) is such that I(reiO) I e-h(8)r -+ 0 uniformly as r -+ 00 (h(e) is the supporting function for the polygon J), is the same as for the function F(z). In rarticular, one may represent any exponential polynomial in that form, if as lone takes the smallest convex region containing all the exponents of that exponential polynomial. The role of the function (z) is played by the sum of the exponents of the functions with exponents lying strictly inside the polygon. We shall construct an example showing that if a vertex of the polygon does not enter into the spectrum, then inside the angle corresponding to that vertex there may be an infinite set of roots. EXAMPLE.

Consider a function F(z) represented by the series CXl

F(z)= ~ ane"n"'sin(I-lJn)z n=l

(6.79)

where ~ Ian I <

As the region lone may employ the triangle Iyl ~ I - x, 0; the points OC n ± ioc n • The point z = 1 does not enter into the spectrum of F(z). We shall show that this function can have an infinite set of positive roots. To this end we shall select the coefficients an in the following way. Suppose that the numbers aI' az, ... , an have already been chosen. Consider the function 00.

x;> 0 and the spectrum of F(z) consists

Pm(z) =

'"

~ ane"nzsill (1- (In)z

(6.80)

n=1

in the strip Iyl <; 1. It is easy to see that there exists a number N m such that for x > N ... and outside the exceptional circles of radius t5 < 1 with centers at

300

ROOTS OF EXPONENTIAL SUMS

[CHAP. VI

the roots of the function sin (i - IXm) Z the modulus of the last term of the sum (6.80) exceeds twice the modulus of the sum of the remaining terms. Moreover, one may suppose that the modulus of the last term in that region is larger than unity. Define a positive number a m+ 1 < 2- m - 1e- 2N,. and then an abscissa Nm+l such that for x> N"'+l and outside the exceptional circles of radius J with centers at the roots of the function sin (I - IXm+l) Z the modulus of the last term in the sum P m+l(z) exceeds the doubled modulus of the sum of all the remaining terms, and moreover is larger than unity. We put a m +2 < 2-m-2e-2Nm+l and so forth. For N m < x < 2Nm and Iyl " I, but outside the exceptional circles, we will have

and 00

!

~

00

aneanZSin(I-lXn)z!<e2Nm

.~ lanl <

21m'

n ,m+l

n=m+l

From the last inequality it follows that in the indicated region the term ame"""· sin (I - IXm) Z of the sum (6.80) exceeds in modulus the entire remaining sum. From Rouche's theorem one finds that the function F(z) has in each exceptional circle of the rectangle N m < X < 2Nm' Iyl ~ 1 exactly one root. Since F(z) is real it follows that that root is real. This example shows that if the conjugate diagram I of the function F(z), with which Theorem 8 is concerned, is a polygon and not all the vertices of this polygon enter into the spectrum of the function F(z), then that function may have an infinite set of roots outside of angles containing the normals to the sides of I. In view of Theorem 8 this set must have zero density.lO 10

If one denotes by)' I (j

=

I, 2, ... ) the positive roots of the function F(z) so constructed,

then

and if one chooses the points

ott

so that the series

diverges. then also the series 00

\.,_1

.t.A

j:l

l' J

will diverge. This shows that in spite of the zero density. the set of roots outside angles con taining the nonnals to the sides of I may be quite "thick ...

Sec. 7]

GROWTH OF A FUNCTION OF CLASS E [

301

7. Growth of a Function of Class E[ Outside an Arbitrary Norming Region 1 In this section we study the general case of an entire function of class E[o for which the conjugate diagram coincides with the smallest convex region containing the entire spectrum. We have shown in § 5 that this always takes place if the norming region 1 is a polygon or if the function is represented by the series (6.01). Moreover, the example of § 5 shows that in the general case this is not the case. The growth of a function of class E[o having that property is characterized by the following theorem. THEOREM 10. If the conjugate diagram of an entire function of class E[ coincides with the smallest cont'ex region containing the entire spectrum of the function, then that function has completely regular growth. PROOF. From the representation (6.63) it follows that a direction 0 which corresponds to a point of the spectrum of the function F(z) is either a point of the spectrum or a condensation point for points of the spectrum. Using this fact we shall show that a direction corresponding to any extreme point of the conjugate diagram is a direction of completely regular growth. Suppose that the extreme point Av of the conjugate diagram IF is not a corner point of the boundary of I. There corresponds to it a unique direction 00 • If i.o is a point of the spectrum, then its direction is a direction of completely regular growth. Now if i.o does not enter into the spectrum, then it is the limit point for a sequence of points A.j of the spectrum, and the direction 00 is a limiting value for directions OJ corresponding to the points Aj • The directions OJ are directions of completely regular growth, and since such directions form a closed set (Chapter III, corollary to Theorem I), then also 00 is a direction of completely regular growth. Now suppose that i.o is an angular point of I. If A.o enters into the spectrum of the function F(z), then all the directions corresponding to it are directions of completely regular growth. It remains only to consider angular points not entering into the spectrum of the function. First we consider angular points which are not common end points for two rectilinear segments of the boundary of the conjugate diagram. In this case there exists a sequence {A,} of points of the spectrum not belonging to a rectilinear segment of the boundary of 1, which converges to A.o as a limit. In addition, the sequence of corresponding directions {OJ} converges to a direction 00 defining one of the sides of the angle (0., ( 2 ) corresponding to the angular point lo. The directions OJ are directions of completely regular growth, and accordingly the same is true for the direction 60 • This direction is not perpendicular to a rectilinear segment of the boundary, and accordingly h~(Oo) = h~(Oo). Moreover, within the angle (0., OJ corresponding to the angular point Av of the boundary of 1, the indicator h(O) is trigonometric. According to Theorem 7 of Chapter III, F(z) has completely regular growth within this angle.

302

ROOTS OF EXPONENTIAL SUMS

[CHAP. VI

Now suppose that the angular point An of the conjugate diagram I is a common end point of two rectilinear segments of the boundary of I. Then the function F(z) may be represented in the form

F (z) = eA.z [V1 (Ze- i8 .)

+ ,~:.I (ze- + Fl. i9t)

'2 (z)),

where 11'1(=) and 11'2(=) are almost-periodic functions with positive spectra, O. EO;; arg z EO;; O2 is the angle corresponding to the point Ao, and F1,2(Z) is an entire function of exponential type, whose indicator is negative inside that angle. Applying Lemma 13, we find from this representation that F(z) is of completely regular growth inside that angle, corresponding to the point Ac,. Thus, F(z) is of completely regular growth in all directions corresponding to the extreme points of l. The directions corresponding to rectilinear segments of the indicator diagram form no more than a countable set. But the set of directions of completely regular growth is closed (Chapter III, § 1, corollary to Theorem 1), and accordingly F(z) is of completely regular growth in the entire plane. The theorem is proved. REMARK I. From this theorem and Theorem 2 of ~hapter III it follows that the set of roots of the function F(z) has a density within the angle (D, 0) for all values of D and 0 with the exclusion of at most a countable set, and this density ~(O) - ~(O) is equal to the arc length of the boundary of I between the supporting points of the lines /6 and 16. 11 REMARK 2. If the function Hz) satisfies all the conditions of Theorem 10 and in addition all the angular points of I enter into its spectrum, and if the function F1(z) satisfies the condition

IF 1 (re iS) Ie -hp(9)r --+ 0 uniformly in 0 as r -

00,

then

<Jl (z) = F (z)+ PI (z) is a function of completely regular growth with indicator hF(O). Indeed, if 00 is a direction corresponding to an angular point trum of the function F(z), then

lim ~ (re iq ) exp (- Aore iq) = r-+0.10

Ao of the spec-

lim F (re i9 ) exp (-I'ore iq ) = a (;'0) r-+oo

*-

O.

Therefore each such direction is a direction of completely regular growth. These directions (values of the angle 0) are densely distributed on the interval (0,2'17') and accordingly cI>(z) is a function of completely regular growth with indicator h 41(O) = hJo'(O). 11 Regensburger [1] obtained for sums of the form (6.01) the following result. If the points are densely distributed on some convex arc and r~i6~ are the roots of the function f(z) lying between the normals to the end points of the are, then 1:;'= 1 r; 1 = oc. This result of course follows from Theorem 10.

).~

Sec. 7]

GROWTH OF A FUNCTION OF CLASS £1

303

In conclusion we consider one further general example of a function of class £]0 where J is the unit circle, for which the smallest convex region containing the spectrum coincides with the conjugate diagram of the function. EXAMPLE.

Suppose that IX)

F(z)

.

= ~'~~)zn.

(6.81)

n=l

where tp(n) is an almost-periodic function on the group of integers. It is knownll that such a function may be approximated by almost-periodic functions of the form Pm (n)

'" a~",),i)."n • =~

(6.82)

1c=1

Thus, the function associated with the function F(z) in the sense of Borel (the Borel transform) has the form

Therefore it easily follows that all the points eiAk , where AI: is a point of the spectrum of tp(n), are singular points of the function ¥'(z). The points eiAt are points of the spectrum of the function F(z) if one takes as the region I the unit circle. We shall prove that the set of singularities of the function ¥,Ng and :±z=O, I, 2..... (6.83)

For the proof of uniform convergence of this sequence on the whole re~1 axis we refer to the theorem of M. Cartwright, proved in Chapter IV. Since the almost-periodic polynomials p.(z) are entire functions of degree not larger than q < 1T, by that theorem it follows from (6.83) that

I p.(x)-P,(x) I < c(q) a U

See, for example, B. M. Levitan [I].

(-00 <x
(6.84)

304

[CHAP. VI

ROOTS OF EXPONENTIAL SUMS

where c(q) is a constant depending only on q. From (6.84) and from the uniform convergence of the sequence p".(z) on the set of real points it follows that this sequence converges uniformly on the entire real axis, which means that it also converges uniformly in any strip parallel to the real axis. The limit function lP(z) is evidently an entire function of degree less than or equal to q. We represent the function q'(z) in the form (Chapter I, § 21)

cp(z)=

2~1

f

a (u)e"'du,

(6.85)

",

where a(u) is the Borel transform of !p{z), and ~. is the rectangle formed by the lines Re u == ±E and 1m u == ±(q + E) (E > 0). For Izl > I + E the function tp(z) may be represented in the form ,'J(z) I

=~ ~r-I fa(u)e"UZ-n-ldu]=~ r~(U)dU. 21t1 'btl" z - eM n-U

~

~

From this representation it is clear that tp(z) h3.s no singularities outside the region into which the rectangle ~E goes under the mapping z == eU • Since E > 0 is arbitrary, the function ·tp(z) may have singularities only on the arc Izl == I, larg zl <; q < fT. Thus the set of singularities of 1p{z) coincides with the closure of the point spectrum of F(z), from which it follows that the conjugate diagram of the function F(z) is the smallest convex region containing all its point spectr~m. By Theorem 10, F(z) is of completely regular growth. 13 l'This example was considered in the paper of H. Mattison[l]. He obtained the following result: if the points e iAk are distributed densely on some arc 61 <; arg z <; 6, and if z, IIR the roots 0/ F(z) inside the angle (-6" -(1), then l:i l/lz/l = co.

CHAPTER

VII

THE THEOREM OF HI:RMITE-BIEllLER FOR ENTIRE FUNCflONS In the preceding chapters we were occupied with the fundamental question of the regularity of the growth of entire functions and the connection of this regularity with the distribution of the roots of the entire function. In a somewhat different section lie other investigations also connected with the distribution of roots and the establishment of conditions for all the roots of an entire function to belong to some region G. In what follows we shall take as the region G the open or .closed upper half-plane. In the exposition of these investigations we shall make use of certain methods and theorems developed in the preceding chapters, in particular, in Chapters III and V. The basic algebraic fact in this domain is the well-known theorem of HermiteBiehler [1] for polynomials. In order that the polynomial w(z) = P(z) +'Q(z).

(7.00)

where p(z) and Q(z) are real polynomials,l not have any roots in the closed lower half-plane 1m z 0, it is necessary and sufficient that the following condition be satisfied: (1) the polynomials p(z) and Q(z) have only simple real roots, while these roots separate one another, i.e., between two successive roots of one of these polynomials there lies exactly one root of the other; (2) at some point Xo of the real axis

<

Q' (xo)P(xo)-Q(xo>P'(xo> > O.

(7.01)

The fact that the two polynomials satisfy condition (I) will be expressed

by saying that ..the roots. of the polynomials p(z) and Q(z) are interlaced," or that "the polynomials p(z) and Q(z) satisfy the interlaced roots condition." The same terminology will be used for entire functions also. I An analytic function is said to be real if aU the cod&:icDts of its expansion into a power ICries in the independent variable are real, or. put differently. if it takes on real values on the real axis. Evidently every entire function can be represented in the form (7.00). where p(z) and Q(z) are real entire func:tions. lOS

306

THEOREM OF HERMITE-BIEHLER

[CHAP. VIJ

We observe that if the roots of the polynomials P(z) and Q(z) are interlaced, then the expression Q'(x)P(x) - Q(x)P'(x) has the same sign on the whole real axis. It is easy to see that in its direct form the Hermite7Biehler criterion does not carryover to arbitrary entire functions. One may show with simple examples that, for the entire function cn(=) = P(z) + iQ(z), the two conditions of reality and interlacing of the roots of the functions P(z) and Q(z) are neither necessary nor sufficient for the function w(z) to have no roots in the lower or in the upper half-plane. Indeed, suppose for example that tp(z) and 'I'(z) are real polynomials whose roots interlace. Then the roots of the real and imaginary parts of the function II>

(z) = eZ~ (Z)

+ ['Hz)

are also real and interlace. On the other hand, this function is of completely regular growth and its indicator diagram is the segment 0 <; x <; 1 of the real axis (Chapter I, § 20). Thus, inside arbitrarily small angles containing rays of the imaginary axis, there is an infinite set of roots (with density ~ = (217)-1) and accordingly the condition of interlacing of the roots of P(x) and Q(x) is not sufficient for all the roots of the function w(z) to lie in one of the half-planes 1m z ~ 0 or 1m z ~ O. Tn order to show that it also is not necessary, we introduce the following example. The function lI>(z) = r').z(z - I )

has only one root i, and it is easy to verify that the real part z cos ).z - sin ).z has for;' > 1 a pair of complex roots, and for;' < 1 three roots between the two roots of the imaginary part -z sin ).z - cos ;'Z closest to zero. For the solution of certain questions of the theory of automation one needs effective :;;-ileria for all the roots of the function tJ,

F (z) =

til

~

~

i )..

akJz e "

(7.02)

k. ;=0

to lie in the left half-plane. N. G. Cebotarev called functions of type (7.02) quasipolynomials. Quasipolynomials depend only on a finite number of parameters, and therefore it is natural to suppose that for them there exists an effective method of solving this problem. In 1942 N. G. Cebotarev found such an effective criterion for a very special case of polynomials [3]. In another paper N. G. Cebotarev [5], generalizing the Sturm algorithm to quasipolynomials, gave a general principle for solving this problem for artitrary quasipolynomials. However the application of this general principle required a generalization of the Hermite-Biehler theorem to quasipolynomials. L. S. Pontrjagin [1] in 1942 generalized the Hermite-Biehler theorem to quasipolynomials of type p(z, eS ), whereP(z, u) is a polynomial in two variables.

Sec. I]

A REAL MEROMORPHIC FUNCTION

301

The papers of N. G. Cebotarev [2; 3; 4; 5; 6] and of L. S. Pontrjagin [1] awakened interest in this group of questions, and there shortly after appeared a series of papers by Soviet mathematicians in this direction. 2 In these papers the question of carrying the Hermite-Biehler criterion to arbitrary entire functions was completely cleared up. . The investigations connected with the Hermite-Biehler theorem later found a new application. They play an essential role in the study of certain extremal properties of entire functions. These applications will be presented in Chapter

IX. I. Representation of a Real Meromorphic Function Mapping the Upper Half-Plane onto the Upper Half-Plane In carrying over the Hermite-Biehler criterion to arbitrary entire functions, an essential role is played by a particular class of entire functions. This class was introduced and studied by M. G. KreIn in his paper "On a class of entire and meromorphic functions," which was devoted to the extension of the Hurwitz criterion' to entire functions. 3 DEFINITION. The entire function w(z) is said to be a function of class HB if it has no roots in the closed lower half-plane 1m z <; 0, and if

I~ I< 1 for 1m z > 0. III

(z) (z)

4

(7.03)

If w(z) is a polynomial, then condition (1.03) evidently follows from the fact that all the roots of the polynomial lie in the upper half-plane, i.e., in this case eondition (1.03) is superfluous. It follows from the Phragmen-Lindelof theorem that the same circumstance also holds for entire functions of zero degree. For entire functions of arbitrary growth the conditions defining the class HB are independent. We note further that it follows from (1.03) that the function w(z)jio(z) belongs in the upper half-plane to the class A, and since the roots of this function in the upper halfplane coincide with the roots of w(z), therefore every function of class HB is .an entire function of class A. In what follows we will frequently need to use the equivalence of condition '(7.03) for the function w(z) = P(z>+ IQ(z), See the papers of B. Ja. Levin [5], N. N. Melman [1], Ju. I. Nelmark [1]. The greater in the book of N. G. Cebotarev and N. N. Melman [1]. I The definition of this class presented here is due to N. N. Melman [IJ. It is equivalent ., to the definition given earlier by M. G. Krein (see the book ofN .1. Ahiezerand M. G. Krein [ID. :\ 4 Here, as usual, we understand by W(z) the entire function obtained from w(z) by replacing .... the coefficients in its Taylor series by their conjugates. I

;art of these results are presenteti

308

[CHAP. VII

THEOREM OF HERMITE-BIEHLER

where P(z) and Q(z) are real functions, to the condition that the function

o(z)=

Q(z) p (z)

maps the upper half-plane into the upper half-plane. Indeed, if we put II)

(z)

w==(z) II)

and 6 = 6(z), then we have fIJ

=

1 +/8 1-/8'

and the unit circle in the plane of IV corresponds to the upper half-plane in the plane of 6. In Theorem I we give a representation of real meromorphic functions which map the upper half-plane onto the upper half-plane. THEOREM I. In order that the real meromorphic fur.ction lp(z) map the upper half-plane onto the upper half-plane, i.e., in order that the inequality 1m lp(z) 1m z > 0 be satisfied for 1m z ¥: O. it is necessary and sufficient that this function be represented in the forms

y(z)=

cz-aoII'(1 _~)(I _~)-1. z- 6 a" \ b k

0

(7.04)

< ak < b"+I' ±k = 0, I, 2, ... , a_I < 0 < bI and c > o. The prime on the infinite product means that the index k takes on all integer values except zero. where h"

PROOF. First we shall prove the sufficiency. From the interlacing of the quantities ak and b" we easily obtain the convergence of the series

and accordingly also the convergence of the series

Thus the product (7.04) converges uniformly on any bounded closed set not containing the points bl: (±k = 0, I, .•.). For all values of the index k we have

l-..!.. arg --.!!!! 1- .!..

= arg (z - aJ -

arg (z -

bJ,

ble

• This theorem was proved by M. G. KreIn in an unpublished paper. It is also implicitly contained in a paper of N. N. Melman [2].

Sec. I]

309

A REAL MEROMORPHIC FUNCTION

so that this argument is equal to the angle subtended by the segment [bt , at] of the real axis from the point z. Thus, from the equation arg ~ (z)

co

= ~ (arg (z ~ Ok) -

arg (z -

bJI

-<Xl

it follows that (J

< arg~(z) < 'It.

Now we shall prove the necessity. Suppose that the function tp(x) maps the upper half-plane into the upper half-plane, i.e., for 1m z > 0 O~argcjl(z)~'It.

Then the function tp(z) has neither roots nor poles in the upper half-plane. Indeed, in the opposite case, on going around a certain contour lying in the upper half-plane and containing a pole or root of the function tp(z), the increment of the function arg '" (z) will be larger or equal to 271, which is impossible. By the symmetry principle the function tp(z) also has neither roots nor poles in the lower half-plane. Thus all the roots and poles of the function ",(z) lie on the real axis. For 1m z < 0 we have -TC

< argcJi(z) < 0,

and when the point z makes one circuit of any circumference, the increment of the function arg '" (z) does not exceed the number 271 in absolute value. Therefore it follows that all roots and poles of the function ",(z) are simple and that on any interval of the real axis the number of roots of the function ,,(z) differs from the number of its poles by no more than unity, i.e., that the roots and poles of the function interlace. The infinite product

Z-ooII'( 1 -Z)( - 1 -Z -)-1 , z- bo b

lJl(z)=--

Ok

k

(7.04')

made up over the roots {at} and the poles {btl of the function tp(z), converges uniformly on any bounded closed set not containing the points of {btl. The ~ion Cll = Cll(z) maps the upper half-plane of the plane of z into the upper JaaIr-plane of the plane of Cll. The function x(z)= ;(z) •

~(z)

iis entire and has no roots in the entire plane, and larg X (z)1 <; 271 for all values

~the variable z.

Thus the entire function u = In X (z) maps the entire plane of z into the Itrip 11m ul <; 271, so that u =const. From this theorem we can at once make the following remark.

310

[CHAP. VII

THEOREM OF HERMITE-BIEHLER

REMARK. In order that the real meromorphic function IV = 1p(z) may be uniformly approximated in any bounded region by real rational functions with real interlacing roots and poles, it is necessary and sufficient that it map the upper half-plane of the plane of z onto the upper half-plane of the plane6 of IV.

Another representation of real meromorphic functions mapping the upper half-plane into the upper half-plane was given considerably earlier by N. G. Cebotarev [6]. THEOREM 2 (N. G. CEBOTAREV). If the real meromorphicfunction 1p(x) maps the upper half-plane onto the upper half-plane, then its poles ale (±k = 0, I, 2, ... ) are 01/ real and simple, and it may be represented in the form CD

I

I '

y(z)=az+b+ ~ ~ Ak ( - - -ale) ale--z

(7.05)

k=o.

( - 00

where a

~

0, b is real, Ak

~

-< < -< (0), Il

0 (±k

W

= 0,

1,2, ... ) and the series

converges. PROOF. By Theorem I, the function 1p(z) may be represented up to a constant multiplier in the form (7.04), and so is the uniform limit of the sequence , Z -1Zo 'fn(Z)=-z - bo

nn (1 -a/cz-) ( 1 -bz-)/c' 1

-n

which may be represented in the form

Yn(z) =

n Ak,n ~ --+cn ale- z k=-n

(for k < IX and k > OJ we take a k = (0), while A Ie•n > 0 and Cn is a real constant. Taking the derivative at the point zero, we find that the sequence of sums n

~

A;.n

k=-n

•

converges to 1p'(0). On the other hand, it is evident that Ale... tends to the residue Ale of the function 1p(z) at the point ale' Passing to the limit as n -+ 00, we find from the inequality N

~ AI:~ n ~ I\I~(O) le3 -N

(n>- N)

/Ile

• This fact was first established by N. G. Cebotarev [6].

Sec. 2]

GENERALIZATION TO ARBITRARY ENTIRE FUNCTIONS

311

that

..

and accordingly the series

~A"

~ al

a

1:=.

converges. Thus we obtain the convergence of the series on the right side of formula (7.05). Now we shall show that the difference between 1p{z) and the sum of that series is a linear function. To this end we fix a number N and consider for n > N the difference N

, (z) I'n

~

~ A",fI + c' ale - Z fI'

1 --1 ) A le, fI ( ak - z ak -

N
le=-N

Evidently the imaginary part of this function is non-negative for 1m z Passing to the limit we have

> O.

N

Im[~(Z)- ~ ~ Ak(-l-_~)]:;po a,,-z a"

(1m z:;P 0).

k=-N

Thus, the entire real function

maps the upper half-plane onto the upper half-plane, and so, from the lemma of N. G. Cebotarev (Chapter V), it is equal to oz + b (0 ;> 0, 1m b = 0). The theorem is proved. 7 2. GeneraUzation of the Theorem of Hermite-Biehler to Arbitrary Entire Functions From Theorem I it is very simple to obtain a criterion for an entire function :10 belong to the class HB, which is analogoui to the Hermite-Biehler criterion. This criterion, due to N. N. Meiman, is giv'm by the following theorem. THEOREM

3.

Suppose that III (z)

= P (z)

+ lQ(z),

7 Another proof of this theorem is obtained immediately from the following representation 41l a function mapping the upper half-plane into the upper half-plane, due to R. Nevanlinna:

<

CD

; (z)

= az + b +

:*re a(t) is a nondecrcasing function.

f l+tz

-CD

t _ z do (t).

312

THEOREM OF HERMITE-BIEHLER

[CHAP. VII

where P(z) and Q(z) are real entire functions, and suppose that

p (z) = A'''~·) (z -

n

co .'(

00)

z) p,,(.!-) a,,' a" (u (0) = 0)

1-

I:--co

and

ii'

Q(z) = B,fI(·)(z-bo)

(7.06)

:J/"( b·,,) ('[1(0) =

(1 -

0)

1c=-co

are their expansions into infinite products. For the function (I)(Z) to belong to the class HB it is necessary and suffiCient thDt the following conditions be satisfied: (Ot) the roots air and bit of the functions P(z) and Q(z) interlace; (fJ) the entire real functions u(z) and /;(z) and the exponents P.,(z/a,,) and P..(z/b,,) satisfy the condition co

u(z)-'[I(z)+

[P1c(:J- P1c( ~-)] =0;

~

(7.07)

1c=-co

(y) the constants A and B are of the same sign. PROOF OF NECESSITY.

It follows from (7.03) that the function F(z)= ~(Z) CJ)

(z)

maps the upper half-plane onto the unit circle, and accordingly that the real function 6(z) defined by the equation

o(z) =

Q(z) P(z)

F(z)-1

= I /,'(z) + 1

maps the upper half-plane onto the upper half-plane. By Theorem 1 its roots and poles are real and simple, and interlace. Since the functions w(z) and w(z) have no common roots, the functions P(z) and Q(z) also have no common roots. Thus we have proved that the roots of the functions P(z) and Q(z) satisfy condition (Ot). From the representation (7.06) of the functions P(z) and Q(z) it follows that B

6 (z) = if

,VI (.kl» (z), co

w(z) = u(z)-'[I(z)

+

~ [PI:C:J-P1c(~)]' 1c=-a>

where 4>(z) is defined by equation (7.04~). From Theorem I, the function T = 4>(z) maps the upper half-plane onto the upper half-plane. Thus we find that in the entire plane of z

I.arg • I< 8(z) (6)

'It,

Sec. 2]

GENERALIZATION TO ARBITRARY ENTIRE FUNCTIONS

313

so that the entire real function 111

=

~ In ~ ~1) =

co

v (z) -

u (z)

+ ~ [p" (:" )- P" ( :" )] "a-CO

maps the entire plane of z onto the strip 11m wi <; 1f, i.e., W = const. One obtains condition ((J) if one observes that K~O) = O. Since the imaginary parts of O(z) and of <J)(z) are of the same sign, therefore A and B are also of the same sign. PROOF OF SUFFICIENCY. ffconditions (at), (fJ) and (1') are satisfied, the function O(z) coincides up to a positive constant multiplier with the function 4>(z), i.e., it maps the upper half-plane onto the upper half-plane. The function F ( ) _ 1 III (z) z -

+

l-llI(z)

maps the upper half-plane onto the unit circle, i.e., inequality (7.03) is satisfied. The functions P(z) and Q(z), and therefore w(z) and w(z), do not have common roots, and from inequality (7.03) the function w(z) has no roots in the half-plane 1m z <; O. The theorem is proved. Observe that the condition AB > 0 may be replaced by the condition

q

(x)P(x)- Q(x)P' (x)

>0

(- 00

< x < 00).

Indeed, if conditions (at) and (P) are satisfied, the function 8(z) maps the upper half-plane either onto the upper half-plane or onto the lower half-plane. In the first case the function O(x) increases monotonically, and in the second it decreases monotonically. It is easy to see that O'(x) ~ 0 and that the sign of B'(x) coincides with the sign of the expression Q'(x)p(x) - Q(x)P'(x). DEFINITION. An entire function oo(z) having no roots in the open lower half-plane (1m z < 0) and satisfying the condition

~ 1-0

wiD be said to be a function of class RB. If w(z) E ifB, then all the common roots of the functions w(z) and w(z) are ~.

Setting up the canonical product R(z) over these common roots, we find

that CD (z)

= R (z) CD1 (z).

(7.08)

where oo1(z) E HB. Conversely, if the function w(z} is represented in the form (7.08), where oo1(z) E HB, and R(z) is a real function having only real roots, then w(z) E HB. It is worth noting that if the function oo(z) is of integral order, then the functions R(z) and w 1(z) may tum out to have greater growth than .GJ(z). In particular, if w(z) is of normal type, then the functions R(z) and wl(Z) . y turn out to be functions of maximal type.

314

THEOREM OF HERMITE-BIEHLER

[CHAP. VII

We also note that if a sequence of functions of class DB converges uniformly in each finite portion of the plane, then the limit function is also a function of class HB. From Theorem 3 it immediately follows that: THEOREM

3'. In order that the entire function w(z) = P(z) +IQ(z)

be a nontrivialfunction of class UB, it is necessary and sufficient that the folio wing conditions be satisfied: (<<') the junctions P(z) and Q(z) may be represented in the form P(z)=R(z)Pdz) and Q(z)=R(z)Qt(z),

while the roots of the entire functions R(z), P1(z) and Ql(Z) are all real and the roots of P 1(z) and Ql(Z) interlace; (P) in the representations of the functions P1(z) and Ql(Z) (u(O)

= 0),

('V (0) = 0)

(a_ 1 < 0 < bJ the entire realfunctions u(z)andv(z) and the Weierstrass exponents P,,(z/a,,) and P",(z/b,,) satisfy the condition co

u(z)-'V(z)+

~

[P (:1c)-P,,( :JJ =0; k

" .. -co

(y) the constants A and B have the same sign.

Condilion (y) may here be replaced by the condition

Q' (x)P(x)- Q(x)P' (x)"> 0 at any point of the real axis. Sometimes it is more convenient to use other conditions for an entire function to belong to the class HB or lTlJ. Before formulating them we introduce the following concept, due to N. G. Cebotarev [3]. DEFINITION. Two entire real functions P(z) and Q(z) are said to be a real pair, if they have no common roots and if any linear combination /JP(z) + "Q(z) of them with real coefficients /J and " has no complex roots. 8

• We refer to roots not lying on the real axis as complex roots.

Sec. 2]

GENERALIZATION TO ARBITRARY ENTIRE FUNCTIONS

315

THEOREM 4. In order that the entire function w(z) = P(z) + iQ(z) belong to the class HR, it is necessary and sufficient that the functions P(z) and Q(z) form a real pair and that at any point Xo of the real axis the inequality Q'(xo) P(xo)-Q(xo) P' (xo)

>0

hold.

PROOF. From the fact that P(z) and Q(z) form a real pair, it follows that the real function

fJ (z) = Q(z) P(z)

(7.09)

does not take on real values for 1m z '# O. so that it maps the upper half-plane either onto the upper half-plane or onto the lower half-plane. If in addition Q'(xo)P(xo) - Q(xo)P'(xo) > 0, then O'(xo) > 0, and O(z) maps the upper half-plane onto the upper half-plane. Hence it follows that the function

F (z) = ~(z) II)

(z)

(7.10)

maps the upper half-plane onto the unit circle, and we obtain inequality (7.03). Since the functions P(z) and Q(z) have no common roots, w(z) and (!i(z) also have no common roots, and it follows from the inequality (7.03) that w(z) has no roots in the half-plane 1m z <; O. Thus, w(z) e HR. Conversely, if w(z) E HR, then w(z) and w(z) have no common roots, and inequality (7.03) is satisfied. But then the real functions P(z) and Q(z) also have DO common roots, the function O(z) is holomorphic in the upper half-plane and 1m O(z) > 0 for 1m z > O. Thus the function O(z) takes on real values only on the real axis, and, accordingly, the linear combination ",P(z) + vQ(z) with real coefficients can have only real roots. The theorem is proved. DEFINITION. Two entire real functions P(z) and Q(z) will be called a generalized real pair if an arbitrary linear combination of them, f.lP(z) + vQ(z), with real coefficients has no complex roots. In this connection, the functions 1»(z) and Q(z) are permitted to have commen roots . .' From Theorem 4 and representation (7.08) there immediately follows THEOREM 4'.

In order that the entire function w(z) = P(z)+ iQ(z)

be a nontrivial function of class H R, it is necessary and sufficient that the functions p(z) and Q(z)form a generalized real pair and that at some real point the inequality

Q' (xO> P (xo) - Q(xo) P' (xO> > 0 be sati8Jied.

316

THEOREM OF HERMITE-BIEHLER

[CHAI'. VII

Now we shall prove that it is impossible to generalize the Hermite-Biehler theorem to a wider class of entire functions.· Here we restrict ourselves for simplicity to entire functions of finite order. THEOREM

S.

If the entire function offinite order UI

(z)

= P(z)+ IQ(z)

has roots and all of its roots lie in the open upper half-plane, and the roots of the functions P(z) and Q(z) interlace, then w(z) E HB. PROOF.

From the estimate of the entire function iii(z) from below (Chapter I,

§ 8), it follows that outside the exceptional circles the function F(z) = w(z)/w(z) satisfies the inequality InIF(z)I
(Imz> 0),

(7.11)

where p is the order of w(z) (e > 0), and M. is some constant. In the half-plane 1m z > 0 the function F(z) has no singularities, and since on the real axis IF(x)1 = 1, by the maximum principle the inequality (7.11) will be satisfied everywhere in the upper half-plane. On the other hand, the function 8(z) = P(z)/Q(z) may be represented in the form (7.12) /j (z) = e' (.~ (z), where g(z) = aoz" + a1z..- 1 + ... + a.., n '" [p] and all of the coefficients al (j = 0, 1, ... , n - ]) are real, and the function ¥,
F ( ) _ 1 + 10 (z) Z -

I-IO(z)'

then IF(z)1 < 1 for 1m z > 0, and so w(z) E HB. Now suppose that n > O. We have 1 Ie' (.'", (z) F(z)= ( •

+

1 - Ie'

.>+ (z)

(7.13)

From the Caratheodory inequality (Chapter I, § 6, Theorem 8'), it follows for 1m z > 0 that ~nO

c1

r < I~ (z) I < c

where Cl and c. are constants and z = rei' (0 On the other hand, we have

r

ll sIn

e'

< 8 < 11').

Re g (re") ~ Do''' cos nfl. • Tbjs

(7.14)

(7.15)

rcsuIt is due to N. N. Melman (see N. G. Cebotarev and N. N. Melman [ID.

Sec. 3]

REPRESENTATION OF A FUNCTION OF CLASS

HB

317

Comparing (7.13), (7.14) and (7.15) we find that outside arbitrarily small angles containing 'the rays arg z = (k + l)7T/n, the function F(z) is bounded. Moreover, F(z) is bounded on the real axis. Taking into account estimate (7.11), we find from the Phragm6n-LindeUSf principle that F(z) is bounded in the upper half-plane and, applying once again the same principle, we find that IF(z) I < for 1m z > 0, i.e., w(z) E HB.

I

3. Representation of a FUDCtion of Class HB In this section we shall obtain a representation of functions of class RB using special infinite products, which is due to M. G. KreIn [1]. In this connection we shall use the following notations. Iff(z) = "i.:':.o(a"

+ ib,,)z", where a" and b" are real numbers, then co

R/(z) or, put differently, 2Rf(z) 0)

(z)

= k-O ~a~

co

and I/(z)

= ~ b~

(7.16)

k~O

= f(z) + /(z) and 2iJf(z) = /(z) -

/(z). Suppose

= zmefll.lll (I - : ) /"( .:) "~1

(7.17)

Ie

is a decomposition of the function w(z) into a canonical product. If w(z) E RB, then the function F(z)= ~(Z) fII(Z)

is bounded in the upper half-plane and therefore, by the remark to Theorem 2 of Chapter V, belongs in this half-plane to the class A, i.e., the series "i.:'. -GO 11m l/at,,1 converges. By Lemma 4 of Chapter V the product x(z) =

ll(1 -~)(1 _ all~ )-1 a"

(7.18)

k .. l

converges uniformly on every closed bounded set not containing any of the points at". We write x,,(z)

• = II(I-...!.}(I- ~ )-1 It-l

and

all

•

0)"

(Z)

.:then we will have

= Z"'e' II (I (.I)

It-l

all

(7.19)

p ( .)

: )e " ;; . "

• ~ _-I

~,,(Z)IXII(z)1-1= eX P 21[lg(z)+ ~ IPlt {"'!')]. .,. (z) •

all

(7.20)

318

[CHAP. VII

THEOREM OF HERMITE-BIEHLER

From the convergence of the products (7.17) and (7.18) we find that the series r.'k=l/P,.(Z/rxk) converges uniformly in any finite portion of the plane. IPk(z/rxlt ) and u(z) = Rg(z), we represent the funcWriting v(z) = Ig(z) + tion w(z) in the form

r.:=1

(7.21) From the Phragmen-Lindelof principle for the half-plane (Chapter I, § 14) one may easily find that

I~ «z» [Zn (z») -11 -< 1 I W

Z

for

1m Z

> 0.

(7.22)

I

Passing in (7.22) to the limit as n - 00 and comparing with (7.21), we see that the entire real function v(z) satisfies the inequality leiv(Z)1 ~ 1 for 1m z > 0, or, what is the same thing, maps the upper half-plane onto the upper half-plane. From a lemma of N. G. Cebotarev (Chapter V, § 3, Lemma 3) we will have v(z) = 'liZ + <5, where 'II ;;> O. We finally obtain

1m 12k

> 0).

(7.23)

with'll;;> 0 and u(z) a real entire function. Conversely, if the function w(z) is represented in the form (7.23), then

(7.24) and w(z) Krein.lO

E

HR. Thus, we have obtained the following theorem of M. G.

THEOREM 6. In order that the entire function w(z) belong to the class HB, it is necessary and sufficient that it can be represented in the form (7.23).

Functions of class HB have evidently the same representation, the only difference being that 1m (lit ;> 0 and the equality sign is possible. 4. The Hermite-Biehler Theorem for Entire Functions of Exponential Type The generalized criterion of Hermite-Biehler is usually necessary in applications to entire functions of exponential type. For this case it may be given a far 10

See N. I. Ahiezer and M. G. Krein [I].

Sec. 4]

FOR ENTIRE FUNCTIONS OF EXPONENTIAL TYPE

319

more complete formulation than in the general case of an arbitrary function of class H B. In this case the conditions (7.03) and (7.06), which are difficult to verify, may be replaced by conditions imposed on the indicator diagrams of the functions w(z), P(z), and Q(z), which in many cases are easily seen to be satisfied. We shall define the defect d w of the function w(z) of exponential type by the equation

2dw = h ( - -;) - h

(i).

For a function of exponential type and in the class A (and thus in HB) the indicator diagram is symmetric relative to an axis parallel to the real axis (Chapter V, § 5). Evidently the defect is equal to length of the segment which this axis of symmetry cuts off from the imaginary axis. taken with the opposite sign. The function w(z) of exponential type will be called a function of class P if (a) w(z) has no roots in theppen lower half-plane; (b) the defect of the function w(z) is not negative. Functions of class P and with degrees not exceeding (J ~ 0 form a subclass which we shall denote by P a • A function which is real up to a constant multiplier will be called a trivial function of class P. The following lemma is very important. LEMMA J. In order that the entire function w(z) belong to the class P, it is necessary and sufficient that it be of the class DB and of exponential type. PROOF.

Suppose that w(z) w(z)

=

E

P and that its decomposition is

.z'I'eG8+ b

n (1 - :J ellA: co

•

k=l

(1.25)

00

(1m a

>- 0

and

~

1 1m

!k

I

< 00).

A:~1

For the function F(z)

= w(z)/w(z) we obtain the decomposition F (z) = e' (p.+t;) X (z).

(7.26)

;where 00

X (z) =

II (1 -

~) ( 1 -- .tA:) -1.

k=l

and p and q are real numbers. From Lemma 5 of Chapter V the function x(z) is of completely regular growth in the half-plane 0 < 0 < 7T and lim. In I X(re'll) I r-+ co

r

O.

(1.21)

320

THEOREM OF HERMITE-BIEHLER

[CHAP. VII

Therefore it follows that F(z) is also of completely regular growth in the same half-plane and h,(ft) = - p sin 6

< 6 < 1t).

(0

(7.28)

When two functions, one of which is of completely regular growth, ar". multiplied, their indicators are added (Chapter III, Theorem 5); therefore h. (6) = hI» (- 6) - P sin 6.

(7.29)

Putting (J = 1T/2, we see that p = 2d",. Since by hypothesis the defect of the function w(z) is non-negative, therefore p ;;. 0 and we obtain from (7.26)

I~ I~ (Z) w (z)

,

1

1m z

for

> o.

Conversely, if w(z) E llB, then Iw(iy) I <; Iw( -iy)1 (y;;> 0). Accordingly, h",(1Tj2) <; h..( -1Tj2), and if w(z) is of exponential type, then it is of class P. The lemma is proved. REMARK. The product of two functions W 1(Z)W 2(z), each of which is of class P, is a function of class P, while the defect of the product is equal to the sum of the

defects of the factors.

The first assertion immediately follows from the lemma. In order to prove the second we observe that the functions _ "'/(Z) F , (z) -=--

(I = l, 2)

w, (z)

are of completely regular growth and that their indicators are hF ;«() = sin (J. It is sufficient to recall that on multiplication of functions of completely regular growth their indicators are added.

-2d.,

LEMMA 2. Suppose that P(z) and Q(z) are entire functions of exponential type, whose roots are real and interlacing, and suppose that 00

P(z) =

z-eu+ e

•

II (1- a:)e I JI -00

and

+. II ( 00

Q(z) =

Z"e h

1-

•

(1.30)

~) eT;

-00

(0_ 1

are their canonical representations. In order that the equation

< 0 < b1) 00

b-o+ ~(:JI -00

-

~.)=O

(7.31)

Sec. 4)

hold, it is necessary and sufficient that the indicator diagrams coincide. PROOf.

321

FOR ENTIRE FUNCTIONS OF EXPONENTIAL TYPE

0/ these functions

From the representation (7.30) it follows that

bIJ'(I- ...!..)(1...!..)-1 b". a"

Q(Z) =ceaaz - O P(z) z- ao

(7.32) t

where c is a constant and 6 = b - a + l:~.., [(I/bt ) - (I/aJ). Denote by ~(z) the infinite product on the right side of the equation, whose convergence has already been proved (Theorem 1). The real function w = ~(z) maps the upper half-plane of the plane of z either onto the upper or onto the lower half-plane of the plane of w (§ 1). By Caratheodory's theorem for the half-plane (Chapter I, § 6, Theorem 8'), the function ~(z) satisfies the inequalities

1 'It',e.')1< c sin' 8 I ISln8 c1 , - <1d".( ill

(6+0. 'It),

(7.33)

in which Cl and Cll are constants depending on the function ~z). It follows from these inequalities that if 0 '" 0, 1T lim In I ~ (,e"I) I O. r ... co

,

Thus, we find that if 0 '" 0, 1T hq (fJ) = hp (6)

=

+ acos 6.

(7.34)

and in view of the continuity of the indicators the equation (7.34) holds for all values of O. Evidently the condition 6 = 0 is equivalent to the identity equation Ja,.{O) == hQ(O). The lemma is proved. We make the following observation, which follows immediately from the equation (7.34). REMARK. If the roots 0/ the functions p(z) and Q(z) 0/ exponential type interlace, then the indicator diagram 0/ the function Q(z) is obtained from the indicator diagram 0/ the function P(z) by a shift through a vector 6. Evidently lfP(z) and Q(z) are real/unctions, then the vector 6 is real as wel/.

Note that if the roots of a real function of exponential type are real, then it is a trivial function of class P. Now we formulate the fundamental theorem of this section.l l

THEOREM 7. In order that the entire function (O(z) = p(z) + iQ(z) be a _ntrivial function 0/ class P, it is neceUQTy and sufficient that the real/unctions p(z) and Q(z) satisfy the conditions: (A) the functions p(z) and Q(z) are representable in the form

p (z) ~ R (z) PI (z) and Q (z) = R (z) Q1 (z).

(7.35)

II This theorem wu proved simullaneous1y and indepeodently by me [5) and by M. G. Krein• . . tile paper or M. G. KreIn wu not pubIisbed.

322

THEOREM OF HERMITE-BIEHLER

[CHAP. VII

while all the roots of the functions R(z), P1(z) and Ql(Z) are real and the roots of P1(z) and Ql(Z) interlace; (B) the functions P(z) and Q(z) are of exponential type and their indicator diagrams coincide; (C) at some real point Q' (xo) P (xo) -

Q(xo) p' (xo)

> O.

(7.36)

By Lemma I, the property of w(z) belonging to the class P is equivalent to w(z) E HR and its being of exponential type. The conditions for the function w(z) to belong to the class HR were given in Theorem 3'. Conditions (ex') and (y) of Theorem 3' coincide with conditions (A) and (C) of Theorem 5. If w(z) is of exponential type, then the functions P(z) and Q(z) are also of exponential type, while the exponential type of at least one of them is equal to that of the function w(z). Conversely, if P(z) and Q(z) are of exponential type, then w(z) is also of exponential type and the exponential type of w(z) is equal to the larger of the types of the functiqns P(z) and Q(z). This deduction follows immediately from the expression of the exponential type of a function in terms of the coefficients of its expansion into power series (Chapter I, § 2). By Lemma :!' the condition of coincidence of the indicator diagrams for the functions P(z) and Q(z) of exponential type with real interlacing roots is equivalent to the equation PROOF.

or, what is the same thing, to condition (P) of Theorem 3'. Thus, if conditions (A), (B) and (C) are satisfied, then the function w(z) is of exponential type and belongs to the class HR, i.e., it is of class P. Conversely, if w(z) is of class P, then by Lemma I it belongs to HR and, accordingly satisfies conditions (ex'), (P) and (y). Since P(z) and Q(z) are also of exponential type, condition (P) is equivalent to condition (B) and thus conditions (A), (B) and (C) are satisfied. The theorem is proved. We shall now analyze more in detail the connection between conditions (A), (B) and (C) and conditions (a) and (b) which 'define the class P. First of all we make the following remark. If condition (A) is satisfied but the indicator diagrams of the functions P(z) and Q(z) do not coincide, then the function w(z) has roots in both halfplanes 1m z > 0 and 1m z < O. Indeed, from Theorem 5 it is easy to show that if, with condition (A) satisfied, all the roots of the function w(z) belong to one of the half-planes 1m z ;;;. 0 or 1m z <; 0, then either w(z) or w(z) belongs to class HR, or else, what is the same thing, to the class P. But in this case condition (B) must be satisfied,

Sec. 4]

fOR ENTIRE FUNCTIONS OF EXPONENTIAL TYPE

323

which contradicts the hypothesis. So the function w(z) must have roots in both half-planes. Thus the requirements that all the roots of the function w(z) should lie in the upper half-plane and that the roots of P(z) and Q(z) should all be real and interlacing are compatible only in the case when the indicator diagrams of the functions P(z) and Q(z) coincide. If now this condition is satisfied, then frolT' the reality and interlacing of the roots of P(z) and Q(z) it follows that either w(z) or w{z) belongs to the class P. REMARK. If the function w(z) is of zero order, then the indicator diagrams of the functions w(z), P(z) and Q(z) reduce to the single point zero. Thus in this case the property of w(z) belonging to the class P is defined by the requirement that w(z) should not have roots in the open lower half-plane. The indicator diagrams of the functions P(z) and Q(z) in this case coincide, i.e., condition (B) is satisfied. Thus it follows from Theorem 7 that for functions of zero exponential type, reality and interlacing of the roots of the functions P{z) and Q{z), and the condition (C), namely that Q'{x)P(x) - Q(x)P'(x) > 0, are necessary and sufficient conditions for the function w{z) = P(z) + iQ(z) not to have roots in the half-plane 1m z <; O. THEOREM 8. If the defect d", of the entire function w(z) of exponential type is positive, then condition (A) is necessary and sufficient for the function w{z) to belong to the class P. PROOF. If condition (A) is satisfied but condition (B) is not satisfied, then, as stated above, the indicator diagrams of the functions P{z) and Q(z) can be obtained one from the other by a shift by a nonzero vector. But since the indicator diagram of the function w(z) is the smallest convex region containing the indicator diagrams of the functions P{z) and Q(z) (Chapter I, § 21), it must be symmetric with respect to the real axis, since the indicator diagrams of the real functions P(z) and Q(z) are symmetric with respect to the real axis, so that d. = O. Thus, if condition (A) is satisfied and d", > 0, then condition (D) is also satisfied, from which it follows that w(z) E P. We consider two examples of the application of the Hermite-Biehler theorem. EXAMPLE 1. Let _(z)=

"

" i)... ... P,,(z), ~e

"-1

(7.37)

where ~ < As < ... < A.., IAll < A.. and P.,(z) are polynomials. The indicator diagram of this quasipolynomial is a segment of the imaginary axis between the points -i1.. and -~, so that d.

= ~ (A" + AI) > O.

[CHAP. VII

THEOREM OF HERMITE-BIEHLER

324

Let W

(z) = P (z)

+ iQ(z).

It follows from Theorem 8 that the condition that the roots of P(z) and Q(z) interlace is equivalent to the condition that all the roots of the quasipolynomial w(z) lie in the open upper half-plane. EXAMPLE

2. Suppose given the differential equation

y" +q (x)y

+ Ap (x)y =

(p (x)

0

>- 0,

1m q (x) = 0).

(7.38)

Determine two solutions tp(x, A) and 'I'(x, A), satisfying the conditions ~(O,

;,) = cos

~'

'I,

(0, ;,) = sin cr,

~'(O. A)~=C05a.

'HO, A)=-sina,

where oc is any real number and the functions tp(x, A) are entire functions of A, since the coefficients of the equation (7.38) are entire functions of A. Let X(x, A) be some solution of the same equatior., satisfying the condition (h real).

Z'(l. ),)+hZ(l, ),)=0

(7.39)

Then under an appropriate normalization z(x. A)=~(X. A)-W(A)~(X. A).

Writing out in detail the boundary condition (7.39), we obtain !p'(l. A)-W(A)·Y(l. ),)+h(~(l, A)-W(A)~(X, A»)=O

or

c) W A -

If' (I. )..) 'l-' (I. A)

+ h'f (I, )..) + h'i' (I, A.) •

(7.40)

By a well-known Abel formula, the Wronskian formed from the solutions q-(x, A) and 'I'(x, A) does not depend on x, and thus ry'(x. A)4(x, ),)-ry(x. A)~'(X, A)= 1.

Moreover, multiplying the first of the relations

X" +q(x)"/.. +AP (x)"/.. = O.

i" +q(x)1. +Ap (x)x::;: 0 by i and the second by X' and subtracting and integrating from zero to I, we obtain I

f (X"X -

or

I

f Ix (x. A)I ~(x)dx = 0, leo. A)i(o. A)_X/(O. A) "/.. (0. A) = (A-A) f Il(X. AH~p(x)dx o

X"'1)dx+ (A -I)

0

I

o

Sec. 4]

325

FOR ENTIRE FUNCTIONS OF EXPONENTIAL TYPE

and 1

UI

(A~:= ~ (A)

=

JI

I. (x. A) Pp (x) dx

> O.

(7.41)

u

Thus O)().) is a real meromorphic function. mapping the upper half-plane into the upper half-plane. This function can have only real poles. The roots of the equation

y' (l. I,)

+ h'~ (l.

A) = 0

are the characteristic numbers of the problem under the boundary conditions (7.42) Thus in this case (for real h) all the characteristic numbers are real and simple. Since h is any real number. 1p(/. ).) and '1"(/, ).) form a real pair. Evidently, if hI ~ h2 the functions '1'(1, ~) + h1tp'(/, ).) and '1'(/. ).) + h2tp'(l, ).) also form a real pair. For h = P + iq we must have 9 (i,)='1(l. i,)+hy'(l, I,)=~(l, 1.)+p'V(l, i,)+iqY'(l. A).

and from Theorem 4 we find that all the characteristic numbers lie in one of the half-planes 1m > 0 or 1m ). < o. If for complex values of ). the general solution of equation (7.38) has an integrable square on the half-axis (0, 00) with the weighting function p(x), then we may put 1 = 00 in the above discussion. From a theorem of M. G. Krein (Chapter V, Theorem 12), we can obtain more complete information on the distribution of eigenvalues of this boundary problem. From (7.40) for h = 0 and h = 00 we find that qJ(/, ).)/tp(/, ).) and qJ'(/, ),)/ ",'(I. ).) are merom orphic functions mapping the upper half-plane into the upper half-plane. We have ,(I, A)

'i' (I. A) ot' (I. A)

= c!- (I. A) -

,'(I. A)
and by a theorem of N. G. Cebotarev (Theorem 2) we have the decomposition

where lk and 14k are roots of the functions tp(/,1) and ",'(1,1) and Ak and B,. arc positive numbers. By the theorem of M. G. Krein, it follows from the existence of this decomposition that tp(/, l)yJ'(/, 1) is an entire function of exponential type. It is

326

THEOREM OF HERMITE-BIELHER

[CHAP. VII

known l • that almost all the characteristic numbers of the equation (7.38) under conditions (7.42) and real h are positive. Putting A = p.2 in (7.43) and expanding each term into elementary divisors, we obtain for the entire function tp{/, p.1)'!p'(/, p.1) an expansion analogous to (7.43). Again applying the Krein theorem, we conclude that the entire function '!p(/, A)'!p'(/, A) is of order not greater than t and is of normal type. Denoting the numbers of roots of the functions VJ{/, A)'!p'(/, A) and '!p(/, A) + h'!p'(/, A) in the circle IAI <; r by mer), nl\(r), we find that since '!p(/, A)'!p'(/, A) is of completely regular growth there exists a finite limit A I. m (r) 1.1= l m - 1 -(tJ. 0).

,. .... "" ,2-

>

From the interlacing of the roots of '!p(/, A) and '!p'(/, A) we may conclude that there exists a finite limit

See, for example, G. Sansone [1]. The di.iCussion presented here is taken from the paper [4] of M. G. Krein. On the basis of these results, he obtains in his paper a very precise characterization of the spectrum of a boundary prot-Iem. 11 11

CHAPTER

VIII

APPROXIMATION OF ENTIRE FUNCTIONS BY POLYNOMIALS WHOSE ZEROS LIE IN A GIVEN REGIONl The representation of an entire function by a power series shows that any entire function is the limit of a sequence of polynomials which converges uniformly in every bounded domain. If we impose on the polynomials which are to approximate the function uniformly the additional requirement that their zeros belong to a certain set, then the limit functions will form a special class, depending on the set. The first results in this direction are due to Laguerre [~], who gave a complete characterization of the entire functions that can be uniformly approximated by polynomials: J. whose zeros are all positive; II. whose zeros are all real. In case II Laguerre formulated the theorem without proof. A proof of his theorem was later given by P61ya [2]. A more complete investigation of the convergence of sequences of such polynomials was carried out by Lindwart and P61ya [I]. They showed, in particular, that in cases I and II (as well as in more general cases) the uniform convergence of a sequence of polynomials in some disk Izl < R implies its uniform convergence on any bounded subset of the plane. We will follow the method of Lindwart and P61ya in the presentation of these questions. In the third section of this chapter we give an exposition of the theorems on "multiplier sequences" which are closely related to the above set of problems. 1. Functions which can be Approximated by Polynomials, all of Whose Zeros Lie in an Angle LEMMA

1.

A set of polynomials of the form nm

Pm(z)

=

II (1 - _z_) = k= t

a'· m. ''',

t: .' Limits

l+c 1 mZ+ ... '

+ Cnm mz"II'

(8.00)

f

,!ranslator's N~te: ~or a sur:vey of rece~~ work in t~is area see the article by J. Korevaar• of polynomials With restricted zeros,' In "Essays In honor of George P61ya .. Stanford 1962. ' • 327

328

APPROXIMATION OF ENTIRE FUNCTIONS

[CHAP. VIII

satisfying the conditions (8.01) and

I Ck.m 1< M

(k = l, 2 •...• P. P -

I

<. P < p)

(8.02)

constitutes a normalfamily in the plane. I PROOF. In Chapter I (§ 4, Lemma 2) we proved the existence of a constant Ap such that for all values of the variable u one has the inequality vii

u'

1(1- u)e u+ 2

+···

+11 1< eAp lul P

(0 <'P

< p <.p+ I).

From this inequality it is easily seen that all polynomials of the set under consideration satisfy the inequality

where s. J.m

"",

=Va-J

f':!t

k. "'

U=

l, 2 •...• p)

and z is an arbitrary complex number. Expressing the quantities sl.m in terms of the coefficients of the polynomials by means of Newton's formulas, and making use of condition (8.02), we find that for certain constants (/i > 0 and for all polynomials of the set considered one will have the inequality (j= l, 2 •...• p).

Hence, throughout the plane,

I Pm(z) I ~ /pMld+a'121+"'+~ apl .. I"

(8.03)

The boundedness of the set of polynomials {Pm(z)} on every bounded subset of the plane implies that it constitutes a normal family.s LEMMA 2. If a sequence of polynomials {P..(z)} of the form (8.00) satisfies condition (8.01) and converges uniformly in some neighborhood of the point 0, then the sequence converges uniformly in every bounded domain. If for such a sequence the number p in condition (8.01) is not an integer, the genus p of the limit entire function does not exceed [p). If the number p is an integer, the limit function has the form F (z) = e·lt.'cfl (z). (8.04)

where" is a constant andCl>(z) an entirefunetion ofgenus not larger than p - I. • A family of holomorphic functions in a given domain is called normal if every sequence offunc:tions of the family contains a subsequence which converges uniformly on every bounded c:IoIed subset of the domain. • See. for CIUIIIlple. A. I. Marlr.uIevie [I].

Sec. 1]

329

FUNCTIONS APPROXIMATED BY POLYNOMIALS

PROOF. It follows from the convergence of the sequence of polynomials P ...(z) in a neighborhood of zero that the coefficients ci .", (j = 1,2, ... ) of these polynomials converge to the corresponding coefficients C1 of the limit functiol) as m - 00, and hence condition (8.02) is satisfied. By Lemma 1 the sequence {P...(z)} forms a normal family and by a well-known theorem of Stieltjes' its uniform convergence in a neighborhood of zero implies its uniform convergence in every bounded domain. It follows from the estimate (8.03) that the limit entire function is at most of order p and normal type. Furthermore, the zeros al: .... of the polynomial P",(z) in an arbitrary circle tend to the zeros al: of the limit function. Thus

or co

~ la"I- P ~ M,

"sl

and hence the genus of the canonical product, constructed with the zeros of the limit function, does not exceed the number p such that p < p p + 1. When p is not an integer it follows therefore that the limit function F(z) is at most of order p and minimal type. When p is an integer the genus of the canonical product is not larger than p - 1. By Hadamard's theorem the function F(z) differs from its canonical product only by a factor of the form

<

that is, it has the form (8.04). THEoREM

1.

If a sequence ofpolynomitlls {P...(z)}

converges uniformly in a

~ighborhood of the origin to a function not identically equal to zero, and all

UFos of the polynomitlls P...(z) lie in the angle 81 <; arg z <; 8a (18a - 81 1 < 17'), IIta the sequence converges uniformly on every finite subset of the plane, and • limit junction has the form

(8.05) co

~la1l:rl
11:=1

61~arga1l:~62 and 61~-arga~6i'

hOOF. We may assume without loss of generality that the polynomials , .(z) have no zeros in some disk Izl < 6 and that P ",(0) = 1. Otherwise we

• See. for example, A. I. Markulevii! [1).

330

APPROXIMATION OF ENTIRE FUNCTIONS

[CHAP. VIII

could consider, instead of the polynomials P m(z), the polynomials

P..(z) =

1

II (z - a,.. "J P..(z),

Iflk, .. I<~

where a,., .. runs through the zeros of P..(z), Since the zeros of P ..(z) tend to the zeros of the limit f'lnction, the convergence of the sequence {J>m(z)} is equivalent to the convergence ofthe sequence {P..(z)}. Noting that lim J>m(O) :F 0 we find that the sequence {J>m(z)/P..CO)} converges uniformly in a neighborhood of zero. We will therefore assume in the following that P..(z) :F 0 in some neighborhood of zero and that P ..(0) = I. We may also suppose that

It follows from the convergence of the sequence of polynomials Pm(z) = 1 +Cl.mZ+C2,mz9+ ... +cnm,mznlll

in the disk Izl obtain that

that cl,m -+ Cl' and consequently IC1 ,ml Am

ftm

k=1

k=l

< M.

Thus we

l: Re(ak,m)-1 < Il':(ak,m)-1 I < M.

and by virtue of the inequality

la,., ..1-1 sin 61 <;

Re (011:, ..)-1

ft . .

M ~ I ak , m , -I < -1-'s n 01 • k=l

Lemma 2 therefore shows that the sequence {P..(z)} converges uniformly in every bounded domain, and that the limit function F(z) has the form 00

F(z)=ce-' z

ll(l-:). k~1

k

The zeros a,. are limit points of the zeros of the polynomialsP",(z), and hence larg a,.1 <; 17T - 61, We still have to prove that larg al <; i7T - 61, From the expansions

and

we obtain that ft m

00

s, = a

+ k-l ~ ail and

SI

•

tit

= l-l ~ ai.lm.

Sec. I] Here T

S1

FUNCTIONS APPROXIMATED BY POLYNOMIALS

= limm~co

S1 m

and limm~co

=

Ok m

~ «5 1 one has Re {e;Tai:~} ~ 0, we find that N

331

Observing that for -«5 1 ~

Ok'

N

Re / eh ~ ak1} = lim Re { eh ~ ak,1ml k=1

m4co

k=l

nm

<.

lim Re

/e h ~ak.\II} =

and taking the limit as N

-+ 00

Re {ehs11.

k=1

11&4<.0

we obtain co

Re {ehol = Re {ei~sll-- Re

/e h

~ ak!} k=1

>- 0

Thus we conclude that larg 0'1 ~ lrr - b1 , and this completes the proof. We remark that if an entire function has the form (8.05), then it can be represented in the form m

F(z)

=

lim m400

cztl(l- az)>nII(1 -~). m

k-l

~

Thus we obtain the following theorem. THEOREM 2. In order that an entire function F(z) can be uniformly approximated in every bounded domain by polynomials, 0/1 of whose zeros lie in the angle 01 arg z 2 (102 - 011 < 7T), it is necessary and sufficient that the function have the form (8.05).

<

<°

We will now investigate the more delicate case 102 - 011 = 7T. A polynomial which has no zeros in the open lower half-plane will be called an H-polynomial. THEOREM 3. If a sequence {P m(z)} of H-polynomials converges uniformly in a 6-neighborhood of the origin to a function F(z) not identically equal to zero,

. then it converges uniformly in every bounded domain. The limit function F(z) belongs to the class H B, and can be represented in the form

(8.06) -where

co

~lakl-2-0.

k=l

If 01/ zeros of the polynomials P m(z) are real, = O.

then 01/ zeros of F(z) are real'

lIItd 1m {3

• In the case of real zeros this is the theorem of Laguerre and P6lya. For the case of complex This theorem was stated ., polya.

Iftoa. the theorem was proved by N. Obrechkoff [1]. Translator's Note:

332

APPROXIMATION OF ENTIRE FUNCTIONS

PROOF.

Taking 0

Q'" ( z)

[CHAP. VIII

< 0 < " we construct the sequence of polynomials

1+ 'I,,,,Z + '2,,,,Z~+ . . . + '''.., ",z"... = P",(z-la) P", (-Ia) =

Since F(~;a) -:F 0 the sequence {Q",(z)} converges uniformly in the disk Izl < " - a. Hence the coefficient cl ,,,, tends to a limit as m - 00, and we will have an inequality Icl.ml < M for some M > 0 and all m = 1,2, .... It follows from the equation

that

"m

~

"m

a

~ 1all, til + fa I~ -<

k=1

~ a+1mak,tll .!; I ak, til fa 12

+

k-l

=

1m 'I, '" < M.

Thus the polynomials Qm(z) satisfy condition (8.01) with p = 2, and by Lemma 2 the sequence of these polynomials will converge uniformly in every bounded domain. The limit function of this sequence is F(z - ;a)/F( -;0), and hence F(z) must have the form (8.04) with p = 2. The H-polynomials Pm{z) belong to the class HB, and hence the limit function F(z) also belongs to HB. It remains to show that,., ;> O. To prove this we suppose first that all zeros of the polynomials P",(z) are real. We may assume without loss of generality that q = 0 in (8.06). Using the notation

and

"

,..I

2

"m

P'" (0) Pm (0) - r til (0) ~-2 $9.", = p2 (0) = ~ ak. JIl' k=1

m

we will have lim",_ao

s2,m

=

S2'

For any positive integer N

and hence

Thus in this case,., ;> O. We now consider the general case. We set

P", (z) = Q", (z)

+ tR", (z).

The uniform convergence of the sequence {P",(z)} implies the uniform convergence of the sequence {p.(z)}, since p ..(z) = P.(f).

Sec. I]

333

FUNCTIONS APPROXIMATED BY POLYNOMIALS

The equations 2Qm(z)

= Pm (z) + 15m(z)

and 2iRm(z) = Pm (z) - Pm (z)

now imply the uniform convergence of the sequences of polynomials {Q ...(z)} and {Rm(z)}, to functions Q(z) and R(z), say. By the theorem of Hermite and Biehler the zeros of Qm(z) and R",(z) are real, and hence by what has been proved already Q(z) and R(z) have the representations Q(z) =

clzq'e-l,zt+~,.

•

n (1 - a~l) ea~) 00

k~l

-

k

and

Since the function F(z) = Q(z)+ iR(z)

belongs to the class H B, condition (fJ) of Theorem 3 in Chapter VII shows that Yl = Y2· Thus the function F(z) can be represented in the form

F (z) = e-y,z'CPl (z), where Yl ;;;. 0 and 1(Z) is a function of at most order two and of minimal type. Such a representation is unique since it implies that hF(O) = -Yl cos 20, and hence Y1 is uniquely determined. Sa REMARK.

A set of H-polynomials

Pm (z) = Qm (z)

+ IRm(z),

satisfying the conditions IP ",(0)1 ;;;. Co > 0, IP~(O)I <; C1 , IP;;'(O)I <; Co' C1 , C2 are arbitrary positive constants),forms a normal family.

C2

(where

PROOF. Denoting the zeros of the polynomials Q",(z) and R",(z) by and Ykm we obtain

XI:...

,. The requirement of convergence of the sequence of H-polynomials (P,,(z)} in a neighbor-

hood of zero can be replaced by the weaker requirement of the convergence of this sequence on a "sufficiently large set" (see Levin [17]).

334

APPROXIMATION OF ENTIRE FUNCTIONS

[CHAP.

VIII

At least one of these sums will be bounded by the number

M=

2(c~+clh) 2

•

Co

Hence, since the zeros of Qm(z) and Rm(z) alternate, we obtain R",

~Xk'~.-< M. J

It follows from these estimates and Lemma I that the polynomials Qm (z) (z - XI"J-I (z - XS"J-I,

Rm (z) (z - Ylm)-l (z - YIl"J-I

form normal families. From this one immediately obtains the normality of the family {P m(z) }.e . Our conclusion remains valid if lim m _ ex; P!!l(O) = 0 (k = 0, 1, 2, ... ,p - 1), lim infm_ oo IpC,t(O) I > 0, and Ip~+l)(O)1 ~ cl , IpC,:+2l(0)1 ~ C2' Indeed, this result follows from the preceding if instead of the polynomials P m(z) we consider the polynomials

where the Zj (j = 1, 2, ... ,p) are the zeros of P,,.(z) closest to O. The requirement lim infm_ co IpC,:l(O) I > 0 is essential, as is shown by the example of the sequence of H-polynomials {zm}, which does not form a normal family in the plane. We remark also that the result can be generalized by replacing the Hpolynomials with entire functions F(z) of the form (8.06). This follows immediately if one makes use of the formula

~ ~ = Fl2 (0) - F (0) F" (0) •

21

+ ~ a~

1"'2 (0)

1

DEFINITION. We will define the class p. as the class of the entire functions which belong to HB and have a representation of the form

F (z) = e -rrq, (z), with y ;> 0 and fl}{z) an entire function of genus not larger than one. THEOREM 4. In order that an entire function F(z) be the limit of a uniformly convergent sequence of H-polynomials, it is necessary and sufficient that it belong to the class p •. • Translator's Note: The proof becomes simpler if one assumes at first that p ..(O) = l. For a different proof see O. Susz, On seqwnces of polynomials and the distribution of their zeros, BuD. Amer. Math. Soc. .., (1943), 377-383.

Sec. I)

FUNCTIONS APPROXIMATED BY POLYNOMIALS

335

PROOF. The necessity follows immediately from Theorem 3. In order to prove the sufficiency we will construct, for a given function F(z) of class p. and prescribed E > 0 and R > 0, an H-polynomial P(z) such that IF(z) - P(z)1 < E for Izl EO;; R. The representation given in Theorem 6 of Chapter VII for an arbitrary entire function of class HB shows that F(z) can be represented in the form 1

00

F (z) =

czqe-TztHHi'z

IT (I - :n) eRe ~

J ,

"~1

with y ~ 0, 6 real and v ~ O. It follows from this representation that for

Izl

< R and sufficiently large n

IF(z)- czqe-Tzt+an8 JJ (I - :J 1< i, 71

where 71

0n=o+i.. + k-l ~ Re

Every polynomial of the form

71

+

=

akl.

(1-

czq(1 _lZ2)Pn(1 anz)Pnn aZ ) . Pn Pn ~~1 " with Pn a positive integer, is obviously an H-polynomial, and for P.. sufficiently large it will satisfy Pn(z)

Iczqe

n (I - ~) 71

-lZ'+crna

P n (z)

I<-;-

" -1

Combining this inequality with the preceding one we obtain

(I zl

~ R).

This proves the theorem. In the next theorem we establish a relation between membership in the class p. for an entire function and membership for its real and imaginary component. THEOREM

5.

In order that the function w(z)

=

P(z)+ iQ(z)

belong to p., it is necessary and sufficient that w(z) belong to the class HB and fltat P(z) and Q(z) be functions of class p •. PROOF.

If w(z) E p., there exists a sequence of H-polynomials W II

(z)

=

P 71 (z)

+ IQ" (z)

which converges uniformly to oo(z). It is clear that the sequences of real Hpolynomials {p ..(z)} and {Q ..(z)} will converge uniformly to the entire functions

336

APPROXIMATION OF ENTIRE FUNCTIONS

[CHAP. VIII

p(z) and Q(z) which, as a consequence, must belong to p.. Conversely, let w(z) E HB and let co

p (z) =

Cl,-T...

•

H,. II (1 - ,,:) eo;" 1&-1 co

Q(z)

•

= C2,-ToI"+1¥ IT (1 -.!..) e6" hI:

k=1

be functions of class p •. It follows from Theorem 1 in § 1 of Chapter VII that the zeros al: and bl: alternate and that co

11

=

12'

~1-~2 + ~(a~ - h~) =

O.

1

and Qm, .. (z)

III

..

!

k~!

= (I - T~r [I +zn- ,(Pl + ~ ak!)]" II (1- hZJ 1

form a generalized real pair, and for bit> alt the polynomial w ......(z) = P......(z) + ;Q ......(z) is an H-polynomial. It is obvious that one can select a sequence of integers n = nm such that the sequences of polynomials {Pm .....(z)} and {Q........(z)} converge uniformly to P(z) and Q(z) as m - 00. The corresponding sequence {w .......(z)} of H-polynomials will converge uniformly to w(z). Thus w(z) E p • .'. Additional properties of functions of class p. will be established in the next chapter. 2. lbeorems OD CompositioD of Polynomials Later on we will need transformations which take H-polynomials into H-polynomials. Theorems on composition of polynomials and on multiplier sequences will play an essential part there. We present these theorems in the next two sections. THEOREM

6 (SCHUR [lD.

f (x) =

ao

If all zeros of the polynomial

+ alx + ar2 + ... + a.x"

(a .. =F 0)

are real, and all zeros of the polynomial

g(x)=bo+b1 x+bll x 2

+ ... +b"X"

(b" =F 0)

.. Korevaar [1; 2] has studied functiODS that are uniform limits of polynomials with zeros in domains compelled of rays issuing from the origin. He has obtainccl vr:ry sipificant raults. In particular, he has atab6shed that an entire function wbidl is a uniform limit of po1ynomiaIa that have no zeros in an angle of opening less than tr can have arbitrarily rapid growth.

Sec. 2]

331

THEOREMS ON COMPOSITION OF POLYNOMIALS

are real and of the same sign, then al/ zeros of the polynomial P(x) = acPo+ l!olblx+2Ia/Jr'l+ ... +kla,p~k (k = mill (m. n»

are real.

If m <; n here and aoho :F 0, then a/l zeros of P(x) are distinct.

For the proof of this theorem we follow the article of I. Schur. As a preliminary we prove two lemmas. LEMMA

3. If a/l zeros of the real polynomial h(x) = CO+c1 x+ •.. +c,.x" (c.+ 0)

are real. Co :F 0 and c" PROOF.

== 0 (0 O.

C~ -

Rolle's theorem shows that all zeros of the derivative of order p - 1 of the polynomial h(x) are real; one has h(P-1) (x)

== (p

- 1)!c,,_1

+ ~: c"x +

(p

+ 1)!....!I 2'.

Cp+l~-

n!"

+ ... + (_ n p + 1)'. c"x -

pH

.

Forming the expression (8.01) for this polynomial we obtain the inequality

pc~-(p+ l)c,_1CI'+I > o. and the assertion of the lemma is an immediate consequence. In particular, it follows from this lemma that in a polynomial h(x) with Co #: 0 no two consecutive coefficients can be equal to zero. LEMMA

4.

If all zeros of the polynomials !(x)=aO+a1x+ .•• +'atlX"

h (x) = Co + ctx + ... + cnxtl

(Co =F O. cn =1= 0)

are real, then all zeros of the polynomial F(x) = h(D)/(x) = cd/(X)

+ c r (x) + ... + cJtll (x) 1

are real. Here every multiple zero of the polynomial F(x) is also a multiple zero Of the polynomial!(xl.

338

APPROXIMA TlON OF ENTIRE FUNCTIONS PROOF.

[CHAP. VIII

The differential operator h(D) can be written in factored form n

h (D) = en

II (D -

Ilk)'

k=1

where all zeros a. k are real and different from zero. For every polynomial rex) and real a. :F the function e 'Xl"q:(x) tends to zero either as x --+ + 00 or as x --+ - 00. By Rolle's theorem the number of real zeros of the function

°

q>1 (x) = (D-Il)q>(x) = ellXD [e-IIXq> (x»)

is not less than that of the function q'i(x). Since the polynomial f(x) has n real zeros the corresponding polynomial (h(X) of degree n also has n real zeros. Consequently F(x) = h(D)f(x) also has only real zeros. To complete the proof of the lemma we still have to show that if ~ is a multiple zero of the polynomial

4> (x) =

f' (x) -1l!(X),

then it will be a multiple zero of f(x). We expand f(x) and l1>(x) in powers of

x

-~:

f(x) 11> (x) = d 1

-

= do + d1(x - ~) + ... + d,,(x zdn + (2d2 - ~dl) (x - ~) + ... -

~)n,

rxdn (x -

If ~ is a multiple zero of the polynomial l1>(x), then d1 2d2 - rxd1 = 0, and hence di - 2dod 2 = 0.

-

~y.

ruJo =

°

By Lemma 3 this relation can not hold when do :F 0. Thus we have do d1 = 0, d2 = 0, that is, ~ is a multiple zero of the polynomialf(x).

and

= 0,

°

PROOF OF THEOREM 6. We first consider the case ao:F 0, bo:F and m '" n. We may assume without loss of generality that am > 0, b.. > 0, and

that all coefficients bk (k = 0, I, ... , n) are positive. we form the polynomial

F (x) = bo!(x)

+ bIZ!, (x) +

0

••

For arbitrary real z

+bmzt" !Cm,(x)

and we write it in the form F(x) = Po (z)

+- P\~Z) x + P~~Z) x 2 + .. + P':r~Z) X"', 0

where and

p .. (z)=p.la ..bo+(IJ.+ 1)la..+1bIz+ +mlambm_ ..zm-.. (!L= I, 2, m). By Lemma 4 all zeros of F(x) are real. 000

000,

Sec. 2)

THEOREMS ON COMPOSITION OF POLYNOMIALS

339

The polynomials Po(z) and Pt(z) have no common zero, for if they did, the origin would be a multiple zero of the corresponding polynomial F(x) and, by Lemma 4, a multiple zero of/ex), contrary to the assumption that ao yI: O. Thus we have a sequence of polynomials Po (z),

PI (z), ... , Pm (z)

with the following properties: (I) The last element P",(z) of the sequence is a constant different from zero. (2) Two consecutive polynomials of the sequence never vanish for the same real value of z (Lemma 3). (3) If one of the polynomials of the sequence vanishes for a real value of z, then the polynomials on either side of it have values of opposite sign for this partic'ular z (Lemma 3). Thus our sequence of polynomials forms a generalized Sturm sequence. If we denote by v(z) the number of sign changes in the sequence of these polynomials for a given value of z, and by r the number of real zeros of the polynomial Po(z), then we will have r~v(-oo)-v(oo).

am> 0

>

and bk 0 (k = 0, I, ... , m) we find that v(oo) = 0 and m, that is, all zeros of the polynomial Po(zYare real. As we pass through a zero of the polynomial Po(z) the number of sign changes v(z) must, When

v( - 00)

=

in this case, increase by the multiplicity of the zero. Since the polynomials Po(z) and Pt(z) have no common zero the number of sign changes can. not increase by more than one, that is, Po(z) has no multiple zeros.

We now consider the case aobo yI: 0 and m

> n.

If instead of the polynomial

g(x) we consider the polynomial

+ ... +

(I - ex)m-ng(x) = boCa) +b1 (e) X bm(:.)x m, we will have bk for k = 0, I, ... , n, lim bde) = { • -+ 0 0 for k n. All zeros of the polynomial

>

bo(a)ao + lIb1 (e)c t x+ ... +n!bn(a)anxR +

... + m!bm (a) amxtn

will be real, and, taking the limit as e -+ 0, we find that all zeros of the polynomial are real. If one or both of the numbers ao and bo are equal to zero, that is, if I(x) =

XP-/I

(x) and g(x) = X'~I (x),

We form the composition of the polynomials I. (x) = (x - atli (x) and K.(x) = (x -a)' Kl (x),

340

APPROXIMATION OF ENTIRE FUNCTIONS

[CHAP. VIII

thus obtaining a polynomial all of whose zeros are real. Taking the limit as 0 we find that the composition of the polynomials f(x) and g(x) has all its zeros real. From the composition theorem of Schur one easily obtains

E -

THEOREM

7 (M ALO). 7

If all zeros of the polynomial

!(x)=aO+a1x+ ... +a,l xn are real, and all zeros of the polynomial

g(x) = bo+ b1x

+ ... +b",xm

are real and of the same sign, then all zeros of the polynomial

P(x)=a o"0+a 1b,x+ ... +ak"kxk (k=min(m, n» are real. PROOF.

If all zeros of the polynomial

!(x)=aO+a1 x+ ... +allxn are real, then replacing x by I/x we see that all zeros of the polynomial

/t (x) = a" + an __ tx+ ... + OOXfl are real also. Forming the Schur composition of this polynomial and the polynomial (l x)" = I +(7)x+(;)x:!+ ... -t-x", we find that the polynomial

+

!2(x)=a,,+a n _ t nx+a ll _ 2n(n-l)x'l+ ... +aon!x" likewise has only real zeros. Substituting again for x the quantity l/x and dividing by n! we find that the polynomial at a2 x -!+ . . . + Ii! an x n ! a (x ) _- ao-t -1f x + "21

has only real zeros. If we now form the composition of this polynomial and the polynomial g(x), we find that the polynomial P(x) has only real zeros.

3. Multiplier Sequences Let

To, T1' .•. , Tn. '"

(8.08)

be an arbitrary sequence of real numbers, and let r*[p(x)] be the operator which takes the arbitrary polynomial

P(x)=aO-t-alx+a2x'l+ ... +allx" , We will not use Mal6's theorem in the sequel.

Sec. 3)

341

MULTIPLIER SEQUENCES

into the polynomial

r*,p(x)) = loa o + 11a l x + l:1a:lx'~+ ...

+ Inan xn .

The sequence (8.08) is called a multiplier sequence of the first kind if the corresponding r·-operator takes every polynomial whose zeros are real into a polynomial of the same class. It is called a muliiplier sequence of the second kind if the corresponding r·-operator takes every polynomial whose zeros are positive into a polynomial whose zeros are real. These concepts were introduced by P61ya and Schur [3]. Laguerre [2] considered a number of sequences of this type. He proved, in particular, that the sequences I ~

1

(m'> 0)

• ",' ",<",+1)'

,

and

(Iql<

I. q. q4, q9 • ...• qn', ...

I)

are multiplier sequences of the first kind, and that for arbitrary real A and 8 the seq uence cos (A 2{). • .• cos A. cos (A

+ {}).

+

is a multiplier sequence of the second kind. If Yn = n, then r·[p(x)] = xP'(x); hence Rolle's theorem shows that the sequence 0, 1, 2, ... is a multiplier sequence of the first kind. We will establish various properties of multiplier sequences. 1. If Yo, YI' ... , Y n' .•• is a multiplier sequence of the first or second kind, then Yk' Yk+l' Yk+2' .•. , Yk+n' ... (k > 0) is a multiplier sequence of the same kind. Indeed, if all zeros of the polynomial

P(x)=, ao+atx+ ... +a"xn are real (or real and of the same sign), and Yo, YI' ... is a multiplier sequence of the first (or second) kind, then all zeros of the polynomial

x-kr* (xkP(x»)

=

a01k+atlk+1x+ ... +an'Yn+kxn

are real. We will assume in what follows that Yo '" O. 2. If a certain element in a multiplier sequence of the first kind is equal to zero, then all subsequent elements are equal to zero. Indeed, all zeros of the polynomial

r* ((l +

x)n)

=

10+ 11 (

7)x +

12 ( ; ) x 2

+ ... + III Xn

wiII be real. It follows from Lemma 3 that if Y n '" 0 and Y" = 0 for some p (0 +1 < O. On the other hand the zeros of the polynomial

r* ,xl'+! - xp-t] = (lp+1X2 - 111 -1) xp- t will be real. But when Y p-IY,,+1

< 0, this is impossible.

342

APPROXIMATION OF ENTIRE FUNCTIONS

[CHAP. VIII

Thus the relations Yn ~ 0 and Y" = 0 (where 0 < p < n) can not hold at the same time. 3. In a multiplier sequence of the first kind the elements either all have the same sign or they have alternating signs. Indeed, it follows from the reality of the zeros of the polynomial YJ>+lX,,+l - YP __ IX,.-l that Yp--l and YP-t 1 have the same sign. A non-negative multiplier sequence of the first kind will be called a sequence, the corresponding operator a r-operator. 4. A r-operator takes an II-polynomial into an H-polynomial. Indeed, let Q) (z) P(z) +lQ(z)

r-

=

be an H-polynomial. We may assume without loss of generality that w(O) "" 0. 8 By Theorem 2 in § I of Chapter VII the polynomials

P(z)=aO+a1z+ ...

+ a liZ"

and

Q(z) = bo+b1z+ ... +bnz" will form a generalized real pair, and aObl - 0lbo > O. A r-operator takes the generalized real pair P(z), Q(z) into the generalized real pair I'(P(z)) = ioao + "1'1°1Z + ... +inallz·.

+

r (Q(z») = iobo+ ilblZ+ ... inbnz". where YOYl(aohl - a1bo) > O. Hence it follows that f[w(z)] is an H-polynomial. Any half-plane of the form IX < arg z < 1T + IX can be mapped into the lower half-plane (1m z < 0) by a simple rotation, and hence, if all zeros of a polynomial R(z) belong to the angle IX .;;; arg z {J (0 .;;; (J - IX .;;; 1T), then the zeros of the polynomial f[R(z)] will all belong to the same angle. In particular, if all zeros of R(z) lie on a ray arg z = c, then the zeros of the polynomial f[R(z)] will lie on the same ray. A r -operator can be extended to a larger class of functions with the aid of the following definition. Let

<

be an entire function. Then we set 00

l' (f(z») = ~

OkikZk

k=O

whenever the latter series converges. • If this is not the case, we can consider the function W(Z + c) and later take the limit

as E -0.

Sec. 3]

343

MULTIPLIER SEQUENCES

THEOREM 8. A r-operator can be applied to all functions of class p. and it takes functions of class p. into functions of the same class.

PROOF.

A function 00

~ a"z" "=0 of class p. is the limit of a uniformly convergent sequence of H-polynomials UI (z)

=

Applying a r-operator to this sequence we obtain a sequence of H-polynomials n

r

lUI" (z)} = ~ i"U".nzk, k~O

where lim n_ CIl Y~I:.n = Y~I:' It follows from the remark made after Theorem 3 that the polynomials f[wn(z)] form a normal family. The lim:t function F(z) of a uniformly convergent subsequence of polynomials r[~,,-:z)] will belong to the class p.. T.he coefficients in the series expansion of suc~ a function can be obtained by passage to the limit; that is, we have 00

F (z) = ~ ika"Z". ,,~O

Since the limit is unique, the whole sequence r[w..(z)] will converge and F (z) =

r [UI (z»).

An important example of a r -operator was constructed by Jensen. MULTIPUERS OF JENSEN [1]. Suppose that all zeros of the polynomial

P(x)=aO+a1x+ ... +a"x" are real. Forming the composition of this polynomial and the polynomial (l + x)n we obtain from Schur's theorem that ail zeros of the polynomial

i\(x) = aO+a1nx+a9n(n-l)x'l+ ... -+a"n(n-I) ... (n-k+ l)x" are real. Replacing x by x/n we find that the polynomial

+ a x + (I - !) a~x9 +... + (I - ! )(I - !) ... (I -- It 1) atx"

PtI (x) = ao

1

11

has all its zeros real. Thus for any positive integer n the sequence

I, I,

I-~ (I-~)(I-~) 11 ' n n '

... , (I - !) (I -

! )...(I -

11 II

1), 0, 0, .. ·

344

APPROXIMATION OF ENTIRE FUNCTIONS

[CHAP. VIII

is a r-sequence. The corresponding r-operator will be denoted by 1..[P(x»). We will sometimes use the notation p..(x) = 1..[P(x»). The operator I Jf] associates a polynomial of degree n with every entire function. It is easy to see that

lim In (f(z)] = fez).

Indeed, let

00

fez)

= 1c-O ~ atz1c

and

"

1,,(/(z»)=aO+a1z+ ~ 1c= z

(1 - !) ... (1-

It

II

1)a1cz1c .

From the inequality for the coefficients of a power series M(R)

la1cI<:~

(M (R) =

max

I.I~B

If(z)~. k

= 1.

2•... )

we obtain the inequalities

I ~"

(1 - ~ )( I- !) ... (1 -

.

It

II

I) akzk I<: :~~:;

1cam+l

and 00

I~ ~

a~

kl

~ M(R),.,,'+1 """ R"'(R-r)

1c=t,,+l

which are valid for Izi we will have

and

< r.

I~

1c=m+l

Choosing R = 2r and taking m sufficiently large

atz1c

l
Furthermore. for sufficiently large n •

I}: (1 1c-Z

~ )(1 -

! )... (1-

m

It

n

I) a~k -

1c=2

(I z1<: r)

and hence If(z)-/n(/(z)JI<.

~ a~k

for

IZI<:r.

1
Sec. 3]

345

MULTIPLIER SEQUENCES

If the function w(z) E p., it can be written as the limit ofa uniformly convergent sequence of H-polynomia\s

.

III",(Z) = ~ a" ..ZII.

"-0 '

Applying the operator I,. we obtain the sequence ·of H-polynomials I,.[w",(z)]. Taking the limit as m - co we conclude that the polynomials 1,,(III(z»)=aO+a1z+(I-

~)aaZ9+ ...

... +(1- ~ )... (1- n 11 l)a,.z", which converge uniformly to w(z), are H-polynomials. We have thus proved the following theorem. THEOREM

9. In order that afunction 00

III (z) = ~ aJlZ"

"=0 belong to the class p., it is necessary and sufficient that the sequence {l..[w(z)]}, II

I"

1111 (z») = ao+atz+ ~ (1- !)(1- !) ... (1- k n 1) a"zk,

"=2 which converges uniformly to w(z), consist entirely of H-polynomials. I We conclude this chapter by proving the following theorem of O. P61ya and I. Schur [1). • This theorem was proved by Jensen [I) for the case of entire functions whose zeros are real. Writing down the conditions of Hurwitz for the polynomials I.[w(z» we obtain a necessary and sufficient condition for membenhip in the class p. in the form of an infinite set of determinantal inequalities. It is interesting to compare this result with existing extensions of Hurwitz's theorem to entire functions. Grommer [I) proved that for real entire functions [(z) = ~:=o a-zl of genus zero the positivity of all Hurwitz determinants, that is, the determinants of the consecutive principal minon in the infinite matrix

4t

0 40 0

44

~

tit 4t 40 0 ...

a.

0

0

0 0 ... 0 0 ...

40 lis lis 41

,

. is ~.ecessary and sufficient in order that all zeros of the function lie in the left half-plane. F~Jlwara [I) extended this result to entire functions of arbitrary genus. However, it was dISCOvered by M. G. Krein [I] that the results of Grommer and Fujiwara.are incorrect. Krein proved that the positivity of all Hurwitz determinants is necessary and sufficient in order that a real function [(z) can be represented in the form[(z) - E{z&) G(z). where E{z) is an arbitrary real entire function and G(iz) is an arbitrary entire function of class HB.

346

APPROXIMATION OF ENTIRE FUNCTIONS

THEOREM

10.

[CHAP. VIII

In order that the sequence of numbers

'To· 'T t ••••• 'Tn ••..

(8.08)

('To =to 0)

be a r-sequence. it is necessary and sufficientthatlhe series 00

!p{z) = ~ Tit Zk

(8.09)

~kl ""'0

converge in the whole plane. and that the entire function !p(z) can be represented in theform

-»l 00

'¥ (z) = with

(J

ceo

(1

+ xkZ)

(8.10)

> 0, K" > 0 and L:'=l K" < 00.

PROOF. Let (8.08) be a r-sequence. operator to the polynomial (1

Applying the corresponding r-

+x)"= ~" (~)Xk, k=O

we find that the polynomial

"

~ 'Tk(~)xk

Qn(Z) =

(8.11)

k=O

has all its zeros real and negative. Putting x = z/n we find that the polynomial P 11 (z)

"

= ~ ~ (I - ~) ... (1 -

k n

1) Zk'

(8.12)

k," 0

has all its zeros real and negative. We will show that the sequence of polynomials {P,,(z)} converges uniformly to an entire function. Indeed, our polynomials will form a normal family since all their zeros lie in the half-plane 1m z 0, while their first three coefficients are bounded and Yo #= 0 (see the remark following Theorem 3). Since the various coefficients in the polynomials P,,(z) tend to limits as n - 00 our normal family has the unique limit function

>

00

'¥(Z) =

~ ~I zlc,

o and hence the sequence of polynomials {P,,(z)} converges uniformly to !p{z). Since the polynomials {P ,,(z)} have all their zeros negative real, Theorem 1 shows that the limit function
R111 () Z =

~ Tk... Ii: ~ """""it Z k-O

Sec. 3]

MULTIPLIER SEQUENCE'S

347

whose zeros are negative real and which converge uniformly to the function cp(z). We form the Schur composition of the polynomials Rm(z) and an arbitrary polynomial

P(z)=aO+a1z+a2z'J+ ... +anz" which has all its zeros real. By Schur's theorem all zeros of the polynomials Sm (z)

=

"

~ ak'Yk mzk

k~O

will be real. Passing to the limit as m polynomial

•

CX)

we find tha t all zeros of the

will be real. Hence it follows that the sequence of numbers (8.08) is a sequence.

r-

REMARK. All alternating multiplier sequences of the first kind can be obtained by forming the series expansions of the functions of the form cp( - z). The proof of the following theorem of G. P61ya and J. Schur [1] is completely analogous to that of Theorem 10. THEOREM

11.

In order that the sequence of numbers

'Yo. 'Yl' ••. , "(,,, ••• be a multiplier sequence of the second kind, it is necessary and sufficient that the series

converge in the whole plane. and that cp(z) be a (constant multiple of a) real function of class p •.

CHAPTER

IX

OPERATORS PRESERVING INEQUALITIES AMONG ENTIRE FUNCTIONS The classes of entire functions considered in Chapters VII and VIII play an important role in the study of inequalities among entire functions and extremal properties of entire functions. S. N. Bernstein discovered the following significant property of entire functions of exponential type [7]. If the entire function fez) of exponential type (] is l!ounded on the real axis, i.e., If(x)l< M

<

( - LX)

< x

(-00

< x < 00).

then

LX).

This theorem, which plays an important role in the theory of approximation, aroused great interest. It was proved again several times and was generalized in various directions. The inequalities in the formulation of this theorem may be written in the form 1f(x) I < I Meig:c I and I/' (x) I <'I (Meioz)'-I. i.e., the Bernstein theorem consists in the statement that under the indicated conditions an inequality between certain functions implies an inequality between their derivatives. Bernstein generalized his theorem [8J, replacing the function e iC1Z by the more general functions w(z) = eiaZq:;(z), where q:;(z) is an entire function of genus zero, having no roots in one of the half-planes 1m z < 0, 1m z > 0, at which ~fll/lm akl < 00. N. I. Ahiezer [2J carried out a further generalization, replacing the function cp(z) by functions of exponential type satisfying certain regularity conditions. In the papers [6; 7J1 these regularity requirements were removed, and it was proved that if w(z) E P is an entire function of exponential type not larger than that of w(z), then from the inequality If(x)I
(-oo<x
1 The bracketed numbers without further indication of an author refer to papers by the author of this book. 348

349

INTRODUCTION

there follows the inequality

If' (x) I -< I w' (x) I

(-oo<x
Essentially one can not further extend the class of functions w(x) playing the same role as eifl% in the Bern~teln inequality. It was proved [7] that if w(z) E P and ('I(Z) E P, then there exists an entire function of lower genus than w(z) which on the real axis satisfies the inequality II(x) I <; Iw(x)l, and is also such that at some point of the real axis If'{x)1 > Iw'(x)l. In the proofs of these theorems essential use was made of the fact that, under the conditions of the theorem, from the inequality I/(x)1 <; lw(x)1 on the real axis and (1OJ <; (1/ there follows the inequality in the half-plane:

If(z) I -< I w(z) I

(Imz-< 0).

N. N. Meiman [2J proposed a direct definition of the majorizing condition by means of the inequalities

If (z) I < I w(z) I,

If (z) I-< Iw (z) I

(Imz -< 0)

(9.00)

and proved that if w(z) E HB there follows from these inequalities the following inequality on the real axis:

If' (x) I -< Iw' (x) I

(-00

< x < 00).

I t remains to note that in this case, in contrast to the preceding, the majorizing relation holding for functions does not necessarily hold also for the derivatives. Therefore in the formulation of this theorem one can not replace the operation of differentiation by multiple differentiation. It is natural to pose the question as to the widest class of majorants in this new sense which preserve the majorizing relation (9.00) after differentiation. It turned out [9J that the widest class of majorants, invariant under multiplication by the function en for any real value of the number., and preserving the majorizing condition (9.00) after differentiation, coincides with p •. In the papers [7; 8J. it was established that the differentiation operator may be replaced in this range of questions by a set of other operators of considerably more general form. In order to obtain these theorems one introduces the concepts of admissible classes of functions and of tJ-operators. A subclass T of the class jjlJ will be called admissible if, from the condition that w(z) belongs to the class T and from inequality (9.00), it follows that w(z) + j(z) E T and I~W(Z) E T (va constant). The function w(z) here is called a T-majorant of the function fez). The general idea of the proof of inequalities for entire functions consists of proving the following theorem. 1fT is an admissible class and iffor Ivl I theJunc!ionj(z) - vw(z) E T, then w(z) is a T-majorant of the entire function j(z). Now suppose that w(z) is a T-majorant of the entire function j(z). Let /(ff] be a linear operator defined on the linear envelope of the clau T and mappillg

>

350

OPERATORS PRESERVING INEQUALITIES

[CHAP. IX

T into itself. Since T is an admissible class, therefore fez) - vw(z) E T for Ivl ;> I. Applying the operator KCf], we obtain KCf(z)] - t'K[w(z)] E T for Ivl ;> I, and, accordingly, Klw(z)1 is a T-majorant for KCf(z)]. Thus the application of such an operator preserves T-majorizing. Linear operators mapping the admissible class T onto itself will be called fBT-operators. The fB ,I- and fB,I.-operators are of particular interest. These will be abbreviated by fB and fB*. In particular, the differentiation operator is a fB- and a !B*-operator. In § 7 of this chapter we shall find the general form of fB- andfB*- operators under certain not very restrictive conditions relating to the continuity of the operator (see [10; 15]). From this general form one obtains, in particular, the fact that any !Boperator which commutes with the differentiation operator has the form

1 !B If(z)]

=

co

~

akjk) (z).

k =0

where 00

F(z)

=

~

akzk

k=O

is an entire function of class P*. The operator rCf) defined in Chapter VIII is the most general form of fB (and !B*)-operator commuting with the operator z(d/dz).

The general expression for the operators fB and ~* required the introduction of the classes P, P*, and HB for functions of several variables 2 [8); fB-operators were defined for classes of functions of several variables and the theorems on the preservation of inequalities were extended. The extension of the theory to functions of several variables made it possible to generalize to such functions the well-known theorem of Laguerre-P6lya given in Chapter VIII (see [8; 9; 15]). These results are discussed in § 5. 1. Majoraots aod Admissible Classes DEfiNITION 1. The function w(z) is called an HB-majorant of the entire functionf(z), if w(z) E DB and for 1m z <; 0

If(z)I~IQ)(z)l,

If(z)I~IQ)(z)l.

(9.00)

If (I)(z) belongs to some subclass T of the class DB, admitting multiplication by a constant, S then we will call it a T-majorant of the function fez). Evidently if w(z) is a T-majorant of the functionf(z), then for lui ~ I also uw(z) is a T-majorant of the same function. I The first Bemlteln inequality for entire functions was carried over to functions of several variables by Bemlteln himself [3J. • I.e., if w(z) E Tthen also cw(z) E T.

Sec. I]

351

MAJORANTS AND ADMISSABLE CLASSES

THEOREM

1.«

Suppose that feZ) and w(z) are entire functions. In order that

the function

'II" (z) = / (z) - 'VIII (z)

belong to the class DB for all v, taken outside the unit circle (Ivl ;> I), it is necessary and sufficient that w(z) be an lTlJ-majorant for fez). PROOF.

First we prove the necessity. Suppose that f/',,(z)

E

nB. Then from

the equation III (z) = -

'V-l~" (z)

lim

> 1 the

it follows that w(z) E llB. Moreover, for Ivl roots in the lower half-plane and accordingly fez) -'V-=I= (z)

function tp,,(z) has no

0

III

for 1m z

< 0 and Ivl > 1.

In other words,

I I~ 1 fez) (z)

III

""

(9.01)

for 1m z <; O. In order to obtain the second majorizing condition, we observe that if 1m z <; 0 1j(z) - 'VIII(Z) I If(z)- 'VIII(Z) I.

<

or, taking into account (9.01) and the fact that w(z) belongs to the class HB,

If(z) 1<(1 +21'VJ)IIII(z)l· Thus, in view of the equality I/(z)I = If(z)1 we shall have throughout the lower half-plane ('for 1'VI ~ 1). From (9.01) we have for -

00

< x < 00

Ilex) I~ III

(x)

""

1

and by the Phragmen-Linde16f theorem (ChfLpter I, § 14)

Ifez) I III

i.e.,

(z)

~ 1

< 0). (1m Z -< 0). (1m Z

Thus w(z) is an ilB-majorant of the function fez). • This theorem was proved by the author for functions of exponential type and formulated as a general principle for the proof of inequalities of the BemAteln type (see [7]. Lemma 3). 1be proof of the generalized inequality of BernAteln with the use of this lemma is an improvement of a method due to N. I. Ahiezer [2]. For the class IIJj the sufficiency pan of the theorem was proved by N. N. Melman 12].

352

OPERATORS PRESERVING INEQUALITIES

[CHAP. IX

Now we prove the sufficiency. If w(z) is an RB-majorant of the function > 1 the function rpf)(z) can not vanish in the half-plane 1m z < O. Moreover, for Ivl > 1 and 1m z < 0

fez), then for 1/)1

I

CP,,(z)/=/f(i)-IIU1(Z)/_lf(z) _vUl(Z)\.\f(Z) -vi ~ 1 + f (z) -

cp" (z)

1IU1

(Z)

-

UI

(z)

UI

(z)

.

UI

(z)

""

Ivi -

Ivl

1.

For real z this ratio is equal to unity, and applying the Phragmen-Lindelof theorem to the function ,,,(Z) cpt) lZ) ,

we find that

1!fI" (z) 1 = 1if" (z) I <.I!fIII (z) I Thus, rpv(z)

E

HB for It'l ;>

(1m z

<. 0).

1. The lemma is proved.

DEFINITION 2. We shall call a subclass T of thl.: class HB admissible if from the fact that w(z) is a T-majorant of the entire fwnctionf(z) it follows that

f(z)+w(z)E T.

From Theorem 1 we see that the class HB is admissible. The class P is also an admissible class. Indeed, if w(z) is a P-majorant of the function fez), then fez) is of exponential type. The functionf(z) + w(z) is also of exponential type. By Theorem 1 this function belongs to HB and accordingly to P (Chapter VII, Lemma I). We shall prove the admissibility of the class p. somewhat further on.

or the Class p.

2. Some Properties

Suppose that P,.(z) is any H-polynomial. From the formula I

Pn(z) Pn(z)

n

~

1

= k=1 ~Z-Zic

in which the z.. are the roots of the polynomial P ,.(z) (1m z.. ;> 0), it follows that for 1m z < 0 P,.(Z)

1m Pn(z)

> o.

Thus P~(z) has no roots in the lower half-plane and accordingly is also an H-polynomial. 6 • The foUowing theorem of Gauss follows easily from the considerations just introduc::ed: 1M rools O/IM .r;lJGl;ve 0/11 polylfOmilll lire 1111 s;llIlIud ill I~ SmIIlleSI co_x rqioll colllllini"K 1M roots 0/ 1M polylfOmilll. .

Sec. 2]

SOME PROPERTIES OF THE CLASS

P

353

Approximating a given function w(z) of class p. by a sequence {P,,(z)} of H-polynomials, we find that w'(z) is also a limit of a sequence {P~(z)} of H-polynomials. Thus, the derivative of a function of class p. also belongs to p •.

The following theorem shows that the class p. is in a certain sense the widest class of functions admitting differentiation. THEOREM

2 [9].

If w(z) is an entire function and iffor any real 7' (et2'w (z))' EHB,

(9.02)

then w(z) is afunction of class p •. PROOF. Suppose that w(z) = P(z) + iQ(z). From Theorem 4' of Chapter VII it follows from (9.02) that for all real values of 7' the functions

pI (z)

+ 'tP (z)

and Q' (z)

+ 'tQ (z)

form a generalized real pair. Thus all the roots of the function P'(z) are real, and accordingly the quantity

+ 7'P(z)

I P' (z)

(9.03)

m P(z)

preserves its sign for 1m z > O. In the upper semicircle of each pole this quantity is negative, and thus it is negative in the entire upper half-plane. By Theorem 2 of Chapter VII the meromorphic function P'(z)/P(z) admits a representation

...

~

-=-2TZ+3+

P' (z) p (z)

[I I] z---ale

k= ..

ale'

=

in which 15 and a" (k 1,2, ... ) are real, y ;> 0 and 1:i_1 P(z) may be represented in the form p (z)

=

...

Ce-T"+a.

la"I-2 < 00.

Thus

-•

II (I -~) eG" ,,= .

so that P(z) E p.. Analogously one shows that Q(z) E p •. Since for any realI', v, and 7' all the roots of the function !J. (PI (z)

+ 'tP (z») + . . (Q' (z) + 'tQ (z»)

are real, by dividing by 7' and passing to the limit as 7' -+ 00 we find that P(z) and Q(z) are a generalized real pair. Hence it follows that either w(z) E HB or w(z) E HB, and from Theorem 5 of § 1 of Chapter VIII, either w(z) E p. or w(z) E p •.

For a non real function w(z) the second case is excluded, since w'(z) E

n.

354

OPERATORS PRESERVING INEQUALITIES

[CHAP. IX

Thus oo(z) E p. This theorem shows the close connection between the class p. and the differentiation operation. 8 In the following theorem we shall prove another characteristic property of the class p •. THEOREM 3 [9]. In order that the entire function oo(z) belong to the class p., it is necessary and sufficient that the condition

w(z-io)EHB

(9.04)

be satisfied for any positit'e a. PROOF. The necessity of the condition (9.04) follows from the fact that H-polynomials have this property and that it carries through under the limit process. . For the proof of sufficiency we note first of all that condition (9.04) is satisfied if the number -ia is replaced by any number U of the lower half-plane. We choose an arbitrary number Zo of the lower half-plane and a real number h > 0 and we put u = Zo - f(h/2). Then for 1m z <; 0

Iw (z+zo-

i

I>

~)

1 01

(z +Zu-l~-) I,

and if we put z = -ihj2, then

101 (zo -

ih) I

> 1 (zo) I.

(9.05)

01

The function oo(z - ih) i~ an HB-majorant for the function oo(z). Indeed, in view of (9.05) we will have

100(Z-ih)l> I w(z)! >! w(z) 1

(1m

z-< 0).

From Lemma I it follows that II)

Cz-lh) lh

II)

(z)

EHB '

and passing to the limit as h - 0 we obtain oo'(z) E HB. If oo(z) satisfies the condition (9.04). then erz(I)(z) also satisfies this condition for any real T. Thus from (9.04) it follows that (e~.rw(z)]'

and from Theorem 2 we obtain w(z)

E

EHB,

p •.

, POlya [4) set the followiDg pro\Jlem in 1914: Can ,.., condude thatJ(z) E P- ifJ(z) ancl all its derivatives have only real zeros? A1andcr [1] remukecl in his ctislcrtatioD that a IOIIlCWbat more pneraI c:oujec::ture wu made by Wiman in his 1ecturea at Uppula in 1911. 1'bae problema of POlya aDd W'IJDIID have inIpireclllWlY invcatiptioJll; relermcea are Jiven by Gol'dbeq aDd 0str0vIkii [I). POIya', c:oqjecture wu recently established by HeIIentein ancl ~ [I). Their rault can be stated .. follows: If areal .,,;,. f-:Iion J(z) I11III. two ~ f{z) IIIIIl r(z) __ 0If/y real _ , . " J(z) E P-. It follows that the CODdusioa of 1beonm 2 remains va1icl if (9.02) is npIacecl by the requiJemeDt that ",(">(z) E HB fOl" k - 0, I, 2. In fact, it follows from the theorem of Hellerstein ancl W'illiamson that .c:z) - AP(z) + IIQ(z) E P- (OI" arbitrary A aDd '" aDd this is euiIy _ to imply that w(z) E p •.

Sec. 3]

355

OPERATORS PRESERVING SUBORDINATION

THEOREM

4.

If w(z) is a P*-majorant of the entire function f(z),

then

w(z)+ /(z)EP*, in other words, P* is an admissible class. PROOF. Suppose that w(z) is a P*-majorant Qf the functionf(z). We shall prove that for any h > 0 the function w(z - ih) is a P*-majorant of the function fez - ih). Indeed, from the fact that w(z) is a majorant off(z) it follows that

(9.06) I f,II(~-ih)1 ~ If(+-ih) I for 1m z <: h. In the proof of Theorem 3 we established in (9.05) that for 1m z 0 the function Iw(z - ih)1 increases monotonically7 together with h(h ;> 0). Therefore for 1m z <: - h we will have

<

I w(z- ih)1 ~ I w(z+ih)1 ~ I1
I/(Z-ih) I.

(9.07)

<

It remains to be verified that the inequality (9.07) is satisfied for -h 1m z ~ O. From the monotonicity of the function Iw(z - ih)1 it follows that for -h <;; 1m z <;; 0 1 w(Z-ih)1 ~ i w(z-ih) I· Moreover, in view of (9.06), for 1m z <: h 1 w(z-

ih) 1 ~

1

/(Z-lh)

1

and in view of the last two inequalities and inequality (9.07) (9.08) 1 w(z- ih) I ~ 1/(i-ih)1 for 1m z <; O. The inequalities (9.06) and (9.08) mean that the function w(z - ih) (h > 0) is a P*-majorant of the functionf(z - ih). By Theorem 1 we find that the function

w(z-th)+ /(z-th)EHB for any h ;> 0, and in view of Theorem 3, w(z) proved.

+ fez) E P*.

3. Operators PreserviDg SubordiDatioD

The theorem is

(~T-Operators)

The following important concept, connected with the extremal properties of entire functions, deals with the so-called ~-operators. DEFINITION 3. An additive, homogeneous operator, 8 defined on the linear rrll'elope of the admissible class T, will be denoted by ~T if it carries functions 01 the class T into functions of the class T. 7 This propeny may be obtained immediately. Indeed. it is obvious for an H-polynomial and is preserved through the limiting process. • The operator !8 is said to be additive if !8[/(z) + fP{z» = !8[/(z») + !8(fP{z»). and homogeneous, if !8[c/(z») = c!8[/(z»).

356

OPERATORS PRESERVING INEQUALITIES

[CHAP. IX

From this definition it follows immediately that every power of a ~T operator, and more generally the products of ~T-operators. are ~T-operators. We shall call a ~l,-operator simply a ~-operator, and a ~p.-operator simply a ~*-operator.' THEOREM 5 [8]. If T is an admissible class and w(z) is a T-majorant of the entire function fez), then 'he function ~T[(I)(Z)] is a T-majorant of the function

~T[f(z)].

The proof is based on Theorem 1. Indeed, since the class T is admissible and w(z) is a T-majorant off(z), therefore fj'" (z)

for

Ivl :>

= f (z) -- Vw (z) ET·

1. By definition the ~T-operator ~T(7',,(Z)J = ~TIf(z)J

- vt\T\w(z») E T

(I v 1 ~ I).

It therefore follows from Lemma I that ~T[w(z)] is an HB-majorant of the function ~Trf(z)]. and since ~T[CIJ(Z)] E T, therl..fore ~T[c.o(z)] is a T-majorant of the function ~T[f(z)]. Theorem 5 makes it possible to obtain various inequalities for entire functions. In what follows we shall restrict ourselves to the study of ~- and ~* operators. THEOREM

If (,I(Z) E p* and iffor 1m z <; 0 the inequalities

6.

are satisfied, then also

If(z) I < Iw(z)l,

!f
If(z)I
1f(ki {z) 1< I wlk) (z) I

(9.00) (9.09)

for rm z " 0 and k = 1,2, ... , while w(I)(Z) E p* (more concisely, the relation of P*-majorizing is pre.serz'ed under d{fJerentiation). If w(z) is not a polynomial and if, in addition, there is at least one point zo(lm Zo " 0) at K'hich (O(z) is not equal to zero, and iffor some kin (9.09) the equality sign holds, then

= C1111(Z) +c2w(z), where CI and Cz are complex numbers, while Icll + ICal =

(9.10)

fez)

1.10

PROOF. In § 2 it was shown that the class p. is admissible and that the differentiation operator maps that class onto itself. Thus by Definition 2

differentiation is a

~*-operator.

In order to obtain the first part of the theorem, it suffices to refer to Theorem 5.

I.

• No nontrivial example or a ST-operator is known with Ta wider admissible class thanp·. In (7J a somewhat more general theorem is proved.

Sec. 3]

357

OPERATORS PRESERVING SUBORDINATION

For the proof of the second part of the theorem we observe that if some point zoOm Zo 0) I(I)(k)(zo) I = lj
<

'fI_ (z) = CtI (z) - e-"f(z) vanishes at the point zoo But q:~k)(z) E p. or else is identically equal to zero. Therefore, if 1m Zo then cp<,:)(z) == 0 and Pie-f) (z)

=

ekCtl(Ie-I)

(z)

< 0,

+ c.

The variation of the argument of c.ok - 1(z) on the real axis is not less than 1f, and therefore, for c ,t: 0 and at some points of the real axis, the impossible inequality

Ifle - I ) (x) I > I cJIe-f) (x) I Accordingly, c == O. Integrating and repeating this discussion

will be satisfied. we find that for 1m Zo

<0

fez)

= e"CtI (z).

Now suppose that Zo is real, i.e., the function q;'~k)(z) has a real root. We put 1p(z) = tp~k-l)(Z). Since lp(z) E p., then OD

¥ (z) ~ [ -1 -Re1] -=-21z+3+

+00

JSI

z-~

~

and (9.11)

This quantity can be equal to zero for x == Zo only if lJ is real and all the roots a; are real. If lp(zo) == 0, then one must replace the functions tp~I:)(z) and tp~k-l)(Z) in the relation (9.11) by lf~k-l)(Z) and tp~k-2)(Z); if q;'~k-2)(zo) = 0, then one takes tp~k-2)(Z) and q;~-3)(Z) and so forth. The case tp.(zo) == 0 and tp~(zO> == 0 is impossible. Indeed, in this case we would have in the neighborhood of the point Zo the following decomposition: fez) ... (z)

== e-+(z-z".fJ (z) OF'

Where p ;> 2, and 6(z) is holomorphic at the point Zo and 6(zo) ,t: O. But such a representation is impossible, since in the lower half-plane the function It1 = f(z)/c.o(z) takes on values lying in the unit disc Iwl < I. Thus, one may suppose without loss of generality that tp(zO> ,t: 0 or lp'(zo) '" O. First we consider the case k == 1. If 1/,(zO>I == Ic.o'(zO>I for some real ZOo then one may choose a real number at such that 9'~(zO> == 0, and, from (9.11), the function 9'.(z) is real up to a constant multiplier.

358

OPERATORS PRESERVING INEQUALITIES

[CHAP. IX

Therefore one may choose real constants Yl and Y2 such that we have identically eiTI! (z) - eiTICI) (z) (z),

==

I,

where ,,(z) is a real function of the class P*. We have eiT,!(z) = P(z)+ iQ(z), eiTow (z) = P (z)

+ IS (z),

while P(z), Q(z) and S(z) are real functions. Moreover, I P(~)+ IQ(x) I IP(x)+ is (x) I,

-<

from which it follows that on the real axis

I Q(x)! -
(-00

< x < 00).

Since S(z) has only real roots and all of these are roots of Q(x). the function {) (z)

=

Q (z)

(9.12)

S (z)

is entire and lD(x) I ~ \ (- 00 < x < (0). We shall prove that fJ(z) == const. To this end we observe first that for -1 < t < \ we have w(z) + tj(z) E P*, and by the generalized theorem of Hermite-Biehler (Chapter VII, Theorem 3) the function tQ(z) + S(z) has only real roots. From (9.12) one finds that the function S(z)[{j(z)t + I] has only real roots for -1 < t < 1. But lD(x) I ~ 1, so that the second multiplier has no real roots. Thus more generally it has no roots for -\ < t < \ and accordingly is equal to a constant. So we have eiTI! (z) = P (z)

+ icS (z),

where P(z) + is(z) = ei1'2w(Z) and c is a real constant. We have considered the case k = 1. We shall show that for k have the same representation of the functionj(z). Suppose that (1m Zo

<: 0).

Then Integrating we have eiTIJIk-2)(z) = P.k-2) (z)+ ICS,k-2) (z)+ c1 •

Replacing z by the root Xi of the function S
> 1 we still

Sec. 3)

359

OPERATORS PRESERVING SUBORDINATION

Since at successive roots of S(k-2)(Z) the function p
C1

= O.

Repeating the argument. we arrive again at

eiT1I (z) = P (z)

+ ieS (z)

(lei < 1),

which we may rewrite in the form

. )1+c .( \1-cI (z) = e'
and c2• we find for fez) the

C1

REMARK. Theorem 6 remains true if in it the differentiation operator is replaced by the more general operator

(D+ -d)!(z) =

f' (z) + ,fez)

for some real T. For the proof it suffices to remark that if w(z) EO P*, then also eTZw(z) EO P*, and if w(z) is a majorant of f(z) , then eTZw(z) is a majorant of eTZf(z). We note the following theorem, which in a certain ~ense may be considered as the converse of Theorem 6. THEOREM 7. If the entire function w(z) is such that for any entire function not representable in the form (9. iO) and for any real 'T it follows from

IW(z)l> I/(z)l,

Iw(z)1 > 1/(z)1

(1m

z< 0)

that the inequalities IW'(.Z)+,W(Z)I '>

1f'(~)+'/(:)I'}

(Imz

I w' (z)+ ,W (z) I> I/f (z)+ ,/(z) I hold, then w(z)

E

< 0)

(9.13)

P*.

+

PROOF. It follows from inequalities (9.13) that w'(z) TW(Z) has no roots in the lower half-plane. The function fez) = cw(z) for lei < 1 is not a tUllction of the form (9.10), and since w(z) is a majorant for it, therefor::

w' (z)+ tw(z) I> i c 11;' (z)+ ,-;;;(z)! i.e., w'(z)

+

TW(Z) E HR. Thus, for any reat

(1m z

< 0, I c I < 1),

T

{e1:Zw (z)l' EH B. It therefore follows from Theorem 2 that w(z)

E

P*.

360

OPERATORS PRESERVING INEQUALITIES

[CHAP. IX

Let us consider some other examples of operators. Let w,,(z) be a Hpolynomial and 1m ot <; O. Then r'llw,,(z) E P, and we find that

= ,uD (,-UCilIi (z)J

(D - «) (Cillt (z»

is also an H-polynomial. If the function w(z) E p., then, approximating it by H-polynomials (/I,,(z), applying the operator D - IX. and passing to the limit, we find that D - at is a ~·-operator. Now suppose that

F(z)=ao+a,z+ ... +a"z"+ ... is an entire function. We define an operator F(D) by the equality F (D) If (z»)

= ao+ ad' (z) + ... + a"I") (z) + ....

(9.14)

The operator is defined for all those functions f(z) for whIch the series (9.14) converges. If F(z) is an H-polynomial, then evidently

"

F(D) (f(z)J = ll(D- ak/) (f(z)J k;1

and F(-D) is a !B·-operator. THEOREM

8.

If F(z) and w(z) are of class p. F (z) = ,-r···F 1 (z)and CIl (z)

= ,-r..'CIl, (z),

where F1(z) and w 1(z) are entire functions of genus one, then for YIYZ series F(D)[w(z)] con1'erges and represents a function of class p •.

(9.15)

the

PROOF. From the estimate for the coefficients of series expansion of an entire function of finite order 2 and of normal type (Chapter I, Theorem 2), we have

(9.16) Moreover, setting in the well-known Cauchy inequality

ICIl(It)(Z) I ~ nl

M", (~t I z I>

+ E), we obtain for any Izl <; pCp > 0), any E > 0 and sufficiently > "P._:

R = ,In/2(YI large "

tI

I CIlltl)(Z) I < n~

tI

(2,fi (1~+ef2 t!r.Hll.".

,.1

It follows from (9.16) and (9.17) that

YI • "00 lim

t

D ..aa!")(Z)I

< (4111s>1•

(9.17)

Sec. 3)

361

OPERATORS PRESERVING SUBORDINATION

Thus the uniform convergence of the series (9.14) in any bounded region is proved. For the proof that a function F(D)[w(z)] belongs to the class p. we approximate the function F(z) by the Jensen polynomials

Fn(z)=ao+a1z+ •..

+a1c(1 - !) ... (1 -

k

n l)z1c+ ... + :~ a,.z".

Evidently F..(D)[w(z)] E p.. From the estimates (9.16) and (9.17) it easily follows that the sequence {F..(D)[w(z)]} converges uniformly in any bounded region to the function F(D)[w(z)], which accordingly belongs to p •. REMARK I. The requirement "1". < 1 is essential. In order to prove this, consider the example: 12 F(D) = r YIDI and w(z) = r Ylll • We will have 00

F(D)[CII(Z»)='-T"'~ (-!~TI)"

Hg,,(z).

11-0

where H2n(z) are Hermite polynomials. For z = 0

Hgn (0) = (- 1)"211 1 ·3 .. . (2n -I) and

For YIY2 ;> REMARK

2.

1 the series converges. For Yl = 0 the operator F(D) is a ~·-operator.

It is also easy to prove, using analogous estimates, that for any )'1 > 0 the operator F(D) is a ~-operator [7]. Another example of a ~·-operator is given by the operator r, defined in § 3 of Chapter VIII. Indeed, by Theorem 9 of Chapter VIII a r -operator carries every function of class p. into another function of the same class. Thus, if the entire functions w(z) E p. and fez) satisfy inequalities (9.00), then r[w(z)] is also a majorant of r[f(z)]. The r-operator of Jensen is particularly interesting. When applied to the function 00

aI

(z) = ~ aJrZ1c Ir-O

it yields the polynomial ;" (z) = ao

,.

+ ~ air ( 1-

! )... (1 -

Ir-l

We formulate the result in the form of a theorem. II

See G. P61ya [3].

k n

1) zIr.

362

OPERATORS PRESERVING INEQUALITIES

THEOREM

9.

If the

entire junction.f w(z)

E

[CHAP. IX

p. and f(z) satisfy inequalities

(9.00), then

17,.

<

<

<

(z) I I ;,. (z) I and II,. (i) I 1;11 (z) I (1m z 0). The meaning of this theorem is that the functions f(z) and w(z) are approximated by polynomials with preservation of the subordination relation, while the operation of passing from functions to polynomials is linear.

4. The Class P and Inequalities on the Real Axis In the study of inequalities for entire functions a special role is played by functions of exponential type. This special role is determined by the fact that in this case the majorizing condition essentially comes down to the inequality If(x) I

< I w (x) I

(-oo<x
(9.18)

on the real axis. More precisely, the following lemma holds. LEMMA 1. In order that the entire function w(z) of class P be a majorant for the entire function f(z). it is necessary and sufficient that the inequality

If(x) I

< Iw(x) I

(-00< x< (0)

he satisfied and that the function f(z) be of exponential type or not exceeding the type a of the function (I)(Z) (or ~ a). PROOF. The necessity of these conditions follows directly from the definition of majorization. For the proof of sufficiency we consider the ratio

.~ (z) = f(z) •

w

(9.19)

(z)'

which is obviously a function of exponential type in the half-plane 1m z <;; O. Moreover. 1'I'(x) 1 <;; I for - 00 < x < 00. This function is of exponential type in the half-plane 1m z < 0 and is bounded on the axis. Therefore it is of completely regular growth (Chapter V, Theorem 11) with the indicator h.,(0) = k Isin 01 (-7T ..; 0 ..; 0). Thus, h,(6) = hg> (0)+ k I sin fJ

I.

(9.20)

We shall show that k ..; O. Since w(z) E P, therefore h",(O) ;;> h.,( -0) for 0 ..; 0, and accordingly h",(O) takes on its largest value a for some 00 (-7T ..; 00 ..; 0). If 0 #: 0, 7T, then -7T ..;

h, (fJ o) - a = k I sin 60

I,

and since max hAO) = or ..; a, therefore k ..; O. Suppose that 00 = 0 (or 7T). It follows from (9.20) that for 1m z ..; 0 the indicator diagram of the function f(z) is obtained from the indicator diagram

Sec. 4)

THE CLASS

P

AND INEQUALITIES ON THE REAL AXIS

363

of the function w(z) by a shift by the vector -ki. If in addition k ::> 0, then we would have 7" ;> v' a 2 + k 2, which is impossible. Thus k <; 0, and by the Phragmen-Lindelof principle l1p{z) I <; 1 on the entire lower half-plane. In other words, If (z) I

-< I w (z) I

(Imz ~ 0).

Carrying through the same argument for the function/(z), we find If (z) I

-< I

W

(z)

I·

Thus w(z) is a P-majorant for the functionf(z). The lemma is proved. We have defined in § 3 the concept of a ~-operator carrying the class P into itself. Since P is an admissible class, by Theorem 5 a ~-operator preserves the subordination relation. From Lemma I, we obtain the following theorem, which is fundamental in this group of questions. THEOREM 10. If w(z) is a function of class P and of exponential type a::> 0, and f(z) is an entire function of exponential type 7" ~ a, and ~[f(z)] is any ~ operator, then from the inequality

If(x) I ~ I w (x) I

(- 00 <

x

<

00)

there follows the inequality I~ If(x») 1~ I ~[w(x») 1

while

~[f(z)]

(-00

< x < 00).

is a function of exponential type not exceeding that of the function

~[w(z)].

In § 2 we have proved that differentiation is a ~·-operator. Moreover, under differentiation a function of exponential type goes into a function of the same type. Thus, differentiation is a 58-operator. THEOREM 11. If w(z) is a function of class P and of exponential type a, andfez) an entire function of type 7" <; a, then from the inequality

If(x)I~lw(x)1

(--oo<x
(9.21)

there follows the inequality 1.f!1c) (x) I ~ 1W(1c) (x) I

( - 00

< x < 00).

(9.22)

If at some point of the real axis at which w(x) =F 0 we have the equality sign, then f(z) =

Where

C1

C1 W (z)

+ cgw(z).

and C2 are complex constants, while ICII

+ Ic2 1 =

(9.10) l.

The first part of the theorem is a direct consequence of Theorem 10, and the second is obtained from Theorem 6 if one observes that Pcp· and w(z) is a P*-majorant of the functionf(z).

364

[CHAP. IX

OPERATORS PRESERVING INEQUALITIES

We shall now show that the class P is the widest class of functions w(z) for which the inequality (9.21) implies the inequality (9.22). Before formulating the theorem we note that if for the entire functions fez) and w(z) inequalities (9.22) are satisfied, then the coefficients of the power expansion of fez) do not exceed the corresponding coefficients of the expansion of O)(z), and therefore the proximate order of f
I/(x)

I

and such that at some point

=,

0)

(x)

(- 00 < x

I

of the real axis

Xu

II' (xo) I > I PROOF.

< 00),

0)'

(xo) ,.

Suppose first that w(z) is of exponential type. If w(xu)

~

0, then

the function

I (z) =

ei • (--:II.) 0) (z),

where" is a real number, is of the same proximate order and type as w(z), and

II(x) I = I

(-00

(x)'

0)

< x < 00)

and for sufficiently large"

II' (xo) , > I

0)'

(xo) ,.

Now suppose that w(z) is of exponential type and that the roots of w(z) lie in both half-planes 1m z < 0, 1m z > O. Choose the point Xu so that w(xu) ~ 0 and w'(xo) ~ O. Suppose for definiteness that 1m (w(xo)/w'(xo <; O. Put z-a I(z) = z- (I 0) (z),

»

where have

IX

is a root of the function w(z) in the lower half-plane. 13 Evidently we '/(x)1 =

I

0)

(x) I

(-00

< x < 00).

On the other hand,

II' (x) I = I

0)'

(x)

III - 2i:' \;) 1m

x 1

(I

I·

Since t

Im -X-(I 18

If 1m (w(x.)/w'(x.»

<

O.

> 0, then one has to choose the root

(I

from the lower half-plane.

Sec. 4]

THE CLASS P AND INEQUALITIES ON THE REAL AXIS

we find that

Re

365

[2i w' (xu) 1m _1-1 > o. (xo) xo U)

iZ

Thus the vector 2; 1m (_1_) Xu

(~(X~, is distinct from zero and forms with

-:x w (xOI

the positive direction of the real axis an angle not exceeding a right angle, so that

If' (xQ) I > 101' (xo) I· Suppose that w(z) has no roots in the lower half-plane. In order that neither w(z) nor w(z) belong to the class P, it is necessary that w(z) have at least one

< O.

root in the upper half-plane and a negative defect d",

We put in this case

f (z) = e2i 5.01 (z), where () = -d.,

> O.

Evidently, I/(x) I == Iw(X) I and

== Re (w'(x)/w(x».

Re (f'(x)l!(x» Moreover, we have w(z)

== e-ih n (1

- z/«,,) eae./2t and therefore ao

1m Ill' (x) III

(x)

(<
= - a+ ~

llc :I 1c_l(X-~Ic)

a +TIc

p" + i"" the roots of w(z».

Analogously

The second terms on the right sides of these equalities are positive, so that (x) I 11m 01 III (x)

< 11m I'I(x) (x) ,.

Since also

Re

01 (x) II> (x)

= Re I' (x)

I(x) ,

I 01 (x) I= If (.It) I;

therefore

1/' (x) I > 101' (x) I

(- 00

< x < 00).

The theorem is proved. Now we shall prove the theorem of N. N. Melman, presented in the introduction to this chapter. THEOREM 13. If w(z) is an HB-majorant of the entire function fez), then for all real t'a/ues of x

If' (X) I ~ I oi (x) I·

366

OPERATORS PRESERVING INEQUALITIES

PROOF.

Suppose that (J)(z) E HB and that j(z)

Then by Lemma I the function q",v(z)

I

=

'f., v (z) 'f •• v (z)

t'ei"w(z)

1<

[CHAP. IX

satisfie~onditions

+ j(z) E HB (v

(9.00).

~ I). i.e .•

1

for 1m z > O. and accordingly arg 9 ",Jx) is a monotonically increasing function. Differentiating the function arg q"..,(x) with respect to x. we obtain

1m fP~.v(x) fPa.v(x)

> O.

Without loss of generality we may suppose that if'(x) > 0 (it suffices to multiply j(z) by a multiplier of the form e'-Y, which does not change the modulus of II'(x)l). Moreover. IIrJ(x)1 ~ Ij(x)l. and for r > 1 the vector ,v(x) makes a complete turn as iX varies from 0 to 27T. Therefore one may choose ex so that q'.. ,t'(x) > O. Under this choice we have 1m {rei"w'(x) + f'(x)} > 0, or taking into account the fact that Imf'(x) = - If'(x)l, we find that lew'(x)1 > II'(x)[. This is true for any r > 1 and accordingly

r..

[ 00' (x) I >- [I' (x) ,. N. N. Merman also proved that the equality here holds only for functions of the form (9,10).

5. Classes of Functions of Several Variables In this section we extend the fundamental concepts and the theorems of the preceding sections to entire functions of several variables. We shall restrict ourselves to functions of two variables. since the general case differs only in that the formulas are somewhat more complicated to write. The entire function w(z. u) (i.e., a function represented by a power series, converging for all values of the variables) is said to be ajunction of class HB2 • if: (a) it does not mnishfor 1m z < O. 1m u < 0, (b) for 1m z < 0, 1m u < 0 the following inequalities are satisfied:

[w (z.

u) 1:>

100 (z. u) I.

I w(z.

u) I:> 100 (z.

u) I.

(z. u) [.

I w (z. u) I:> 100 Evidently the limit of a uniformly convergent sequence of functions of class RTJ2 is a function of the same class . . The function w(=. u) is said to be an HB2-majorant of the entire function fez. u) if for 1m z < O. 1m u < 0 Iw(z. u)I:>I/(z.

u)l.

Iw(z.

u)I>-I/(z. u)l.

We shall prove a theorem analogous to Theorem 1.

Sec. 5]

CLASSES OF FUNCTIONS OF SEVERAL VARIABLES

THEOREM I'. that the function

Suppose that w(z, u) and f(z, u) are entire functions.

VUl

367 In order

(z, u)

belong to the class H B2 for all v taken outside the unit circle (Ivl ;> 1), it is necessary and sufficient that w(z, u) be a HB2-majorant for f(z, u).

We first prove the necessity. We have UI (z, u) = lim V-l~" (z, u)

".... ""

so that w(z, u) E HB2 • Moreover, if 1m u <; 0, then the function of z q;iz, u) E HB in the variable z (we shall write in this case 'l'v(z, u) E DB.), and accordingly, from Lemma 1 1 f(z.

u)1 ~ 1UI(Z, u)

l!(z. u)I~IUI(z,

I,

u)l·

Analogously one proves that If(z.

")1 ~ IUI(z.

(9.23)

u)l·

Finally, from the inequality Il(z. u)-v-;;;(z. u)I-
-< 0)

it follows that

(1m

Z

-< O.

1m

u -< 0),

and the function 9(Z. u) = 7(z. u) II>

(z. u)

is bounded in the indicated region. It follows from (9.23) that for real z and for 1m u < 0

19 (z.

u) I

-< 1.

For a fixed u (1m u < 0) we apply the Phragmen-Lindelof principle for the half-plane 1m z ~ 0 to the function 1p{z, u) and obtain (1m z -< O. 1m u -< 0). Or

I!(z. UlI-
368

OPERATORS PRESERVING INEQUALITIES

[CHAP. IX

We now prove the sufficiency. It w(z, u) is an RBI-majorant for /(z, u), then, fixing u (1m u < 0), we see from Lemma 1 that the function fP.(z, u) E HBz• Accordingly, it has no roots for 1m z < 0, 1m u < 0, and

I'fI" (i, u) I '" 1'fie (z, u) I Analogously one proves that

u) I '" 1'fI., (z,

(1m z '" 0, 1m u '" 0).

(9.24) (1m z '" 0, 1m u '" 0). Just as in the proof of Lemma I, we find that for 1m z .; 0, 1m u .; 0, I 'fI., (z,

u) I

(Ivl»). tor a real z and 1m u

< 0 we have, in view of (9.24),

I,,, I'" ) (z. u) ,,, (z. u)

and, applying to this function the Phragmen-Lindelof principle in the halfplane 1m z .; 0, we obtain a third inequality

I'fill (z. U) I'" I'fie (z. (1m Z", O. 1m The theorem is proved. Admissible class and

~-operator

u'"

u)l 0).

are defined as i'l the case of one variable.

THEOREM S'. If w(z, u) is a T-majorant oj the Junction f(z, u), then ~T[W(Z, u)] is a T-majorant of the function ~T[f(z, u)]. The proof does not differ from [hat of Theorem S. DEFINITION. A polynomial not vanishing for 1m z < 0, 1m u < 0 is called an HI-polynomial in the variables z, u. It is easy to see that every H.-polynomial P(z, u) belongs to the class RBI' Indeed, it belongs to HB in each of the variables, under the condition that the other variable is fixed in the lower half-plane. Moreover, for fixed u (1m u .; 0) the function P(z. u)

'fI(z, u)= P(z. u) tends to a finite limit as Izi - 00 and, accordingly, is bounded in the lower half-plane. Since for a fixed real z the polynomial P(z, u) is an H-polynomial in u, therefore IfP(z, u)l .; 1 for 1m U 0 and real z. Applying the PhragmenLindelof principle to this function in the half-plane 1m z 0 for a fixed u (1m u " 0), we obtain I P (i. I = 1.0 (z. u) I

<

u)

'" IP (z.

u) I (1m z", 0. 1m u'" 0).

>

Sec. 5]

CLASSES OF FUNCTIONS OF SEVERAL VARIABLES

369

Evidently a product of H-polynomials of different variables, P1(Z)P2(u), is an H 2-polynomial. Lemma 1 makes it possible to construct nontrivial examples of H2 -polynomials in the following way. Suppose that Q(z, u) is any polynomial of two variables. Suppose that n is the highest degree in which z appears, and that m is the highest degree of u. Then for real z and u the function

I

Q(z. u)

(z - l)n (a - I)'"

I

is bounded. Choosing a number M larger than or equal to the least upper bound of this function for real z and u, we easily find, using the PhragmenLindelOf principle for the half-plane, that the function M(z - O"(u - i)'" is an 1m2-majorant of the polynomial Q(z, u) and that by Lemma 1',

P(z. a)=M(z-l)n(u-l)m+Q(z. u) is an H 2-polynomial. Analogously one may construct nontrivial examples of entire functions of arbitrary growth and of class HBz, taking as majorants the products of functions of class HB and of one variable. Note that a partial derivative of an Hz-polynomial is an Hz-polynomial. Evidently, the class of functions which are uniform limits of Hz-polynomials is invariant under partial differentiation. We define as follows the class of two variables.

P:

DEFINITION. The class P: of functions of two variables is the set of all entire functions w(z, u), satisfying the conditions: 1. w(z, u) E BBz. 2. For fixed u(lm u " 0)

w(z. u)EP:. 3. For fixed z(Im z " 0)

w(z. u)EP~. It is easy to see that P: is an admissible class. Indeed, it follows from Lemma l' that if w(z, u) is a P:-majorant of the functionf(z, u), then

w(z. u)+f(z. u)EHB'J. Moreover, for a fixed u(lm u <; 0), w(z, u) is a P:-majorant of the function· /(z, u), and accordingly (D(z. u)+f(z. u)EP;.

In the same way w (z.

u)+ f(z, a)E P~

(1m z ~ 0).

We note further that, as one sees from Theorem 4 of Chapter VIII, the limit fUnction of a uniformly convergent sequence of Hz-polynomials belongs to the

class p.2·

370

OPERA TORS PRESERVING INEQUALITIES

We shall show that the class Indeed, if for positive hand k

P~

[CHAP. IX

has the same characteristic property as p •.

w(z-lh. u-Ik)EHB'}..

then w(z, u) E HB2 • For a fixed u (1m u <;; 0) (I)(z - ih, u) E HTJ., and from the characteristic property of the class p~ one obtains (I)(z, u) E P~. Analogously one shows that w(z, u) E Conversely, if OI(Z, u) E P~, then evidently for h ~ 0 also 1m Z <;; w(z - ih, u) E Moreover, in view of the characteristic property of the for 1m u <;; we have w(z - ih, u) E We need still to prove that class w(z - ill, u) E HTJ2 • This now comes down to the proof of the inequality

P:.

P:.

P:

°

°

P:.

I w(z-lh, u) I ~ I w(z- th.

u)1

(1m

z <. 0,

1m

U

<. 0).

In order to prove this we note that for an H-polynomial, which means also for a function of class p., the following inequality holds:

I w (z -

2ih) I ~

I W (z) !

for h ~ 0, 1m Z <;; h. Therefore from w(z, u) that for h ~ 0, 1m Z <;; h, 1m u <;; 0,

E

P: (for 1m u <;; 0) it follows

Iw(z-2Ih, u)I~lw(z. u)I~lw(z,

Putting

ZI

=

ih, we find that for 1m

Z -

ZI

<;; 0, 1m u <;;

U>!.

°

IW(Zl-ih, u)!>lw(Zl--ih, u)l. Thus, (I)(z - ih, u) E P!. Carrying out the same operation over u, we find that, for h > 0, k > 0, w(z - ih, u - ik) E P~. We shall prove the following theorem on the representation of functions of class P;. THEOREM

to the class

In order Ihal an entire function (I)(z, u) of 11m rariables belong it is necessary and sufficient thai it be representable in the form

14.

P!,

(9.25) where Yl and 1'2 are non-negafil'e numbers, and w1(z, u) is an entire function in HB2 and of genus one in each of its variables under the condition that the other variable is fixed in the lower half-plane.

PROOF. The sufficiency of this condition for the function w(z, u) to belong to the class P~ follows immediately from the definitions of the classes p. and P;. For the proof of necessity we note that if the function w(z, u) belongs to the class for 1m z ~ 0, it follows from the representation of a function of class HE (Chapter VII, Theorem 6) that the function

P:

'P~ (a) =

...: (z. u) ... (z. u)

Sec. 5]

371

CLASSES OF FUNCTIONS OF SEVERAL VARIABLES

maps the lower half-plane 1m u < 0 onto the upper 1m u > O. Therefore the whole family of functions of u, {q>.(u)}, is normal for 1m u < 0 (z a parameter varying in the lower half-plane). From the decomposition lJI'(u) = -2y(u)z

+

.,,(u)

+

+

A

i

(_1_

k=1

Z -

ak

+ Re (~)) ak

(y(u)

~ 0)

(a k functions of u) it follows that

-1(u)=

lim Z-l
Since the family offunctions {z-llJl.(u)} is normal, the convergence is uniform, and accordingly y(u) is a holomorphic function in the lower half-plane. Since, on the other hand, y(u) takes on non-negative values in the entire lower halfplane, y(u) is a constant. Thus, w (z, u)

=

e- T,Z'w2 (z. u).

where w 2(z, u) is of genus one relative to the variable z for 1m u ~ O. Analogously one separates out the multiplier r1'2 U •• The theorem is proved. DEFINITION. The operator K[f(z)], carrying entire functions into entire functions, will be called continuous if it carries every sequence of entire functions converging uniformly in every finite region into a sequence converging in the same sense. THEOREM 15 [9]. El'ery continuous lB:-operator orer entire functions of one variable z carries functions w(z, u) of class P; into functions of the same class. PROOF. First of all we verify that the continuous operator lB: carries a function of class into an entire function of two variables. Indeed, by the very definition of the operator lB: the function

P;

=~: [w (z. u»)

belongs to P~ for Tm u ~ O. Since for 1m u ~ 0 w(z, u) is a P:-majorant of w(z, Ii) and P: is an admissible class, therefore w(z, u) + w(z, Ii) EP!. Ac* cordingly, _
+ h)-, (z. u) _ (0*[11) (z. U+ h)-II) (z. U)] h

-

"'.

h

> O.

Now we

•

In view of the continuity of the operator lB: it therefore follows that for all u (uniformly in z in a finite region) there exists a limit

'u ,

(z. u) =

•

I

e. (w. (z.

u)l.

and accordingly q;(z, u) is an entire function of u for all z. Hence it follows that rp(z, u) is an entire function of two variables.

372

OPERATORS PRESERVING INEQUALITIES

[CHAP. IX

Now we shall show that it belongs to P~. Since a ~~-operator carries every function of into a function of the same class, therefore for 1m u ~ !p(z, u) E P:, and consequently for 1m z <; 0, 1m u <; 0,

°

P:

1~ (z. u) I ~ 171

(z.

u) I·

In order to obtain the two other inequalities we observe that since for 1m u ~

°

w(z, u) is a P:-majorant for w(z, 11), and a ~~-operator preserves subordination,

therefore

P:

Thus cp(z, u) E HB2 • In view of the characteristic property of the class for • k ~ we have w(z, u - ik) E P~ and consequently q:(z, u - ik) E HB2 Making use of the characteristic property of the class P!, we find that for 1m z ~ 0 we have q(z, u) E Thus conditions 1, 2 and 3 are satisfied and consequently q(z, u) E P~. The theorem is proved. 14

°

P:.

The apparatus developed for '!3-operations makes it now possible to obtain the following generalized theorem of Laguerre-f6lya on functions of several variables. THEOREM 16. In order that the entire function w(z, u) may be uniformly approximated by a sequence of H2-poly"omials in the rariables z and u, it is necessary and sufficient that it belong to the class P~, and accordingly hare the representation (9.25). PROOF. We have already proved that the limit function of a sequence of H 2-polynomials belongs to P~. Now suppose that 00

(I)

(z, u)

=

~ akpzkup k.p~O

is a function of class P;. Applying to it the real ~·-operators of Jensen, In in z and I", in u (Chapter VIII). we obtain a sequence of polynomials n

(L)n,m(z, u) = a oo

+ ~akO( 1-{} .. (1- k ~ I)Zk k~l

m

+ ~aOp(I- ~) ••. (I-P m l)up p~l

n. ,.

+ ~ akP( 1- ~) ... (1- k n 1)(1_ ~) ... (I-Pm 1) ZkUP, k.p~l

Evidently the theorem remains true jf the class p. is replaced by the class HB, and the ~·-operator by a ~ H B-operator. In this case the proof of the theorem becomes Simpler. The continuity of the ~:-operator is used only in proving that it carries an entire function of twO variables into an entire function. We observe that this operator on functions of two variables is also continuous. 14

Sec. 6]

GENERAL FORM OF THE OPERATORS ~ AND ~.

373

which, as n -+ 00 and m -+ 00, converge to the function w(z. u). By Theorem 15 these polynomials are H 2-polynomials. 6. General Form of the Operators ~ and

~.

Suppose that Klf(z)] is a linear continuous operator defined on some class of entire functions of one variable and mapping these functions into entire functions, and suppose that its region of definition V contains the function e- UZ for any fixed u. Evidently.

ep (z, u) = K [e-ut)

(9.26)

is an entire function of two variables. Differentiating both sides of the equation n times with respect to u and putting u equal to zero, we obtain K[zn)=(_l)nep~)(z, 0).

Thus if

f(z)

=

(9.27)

co

l1 a~k k-O

is an entire function belonging to V, then in view of the continuity of the operator co

K[f(z)] == l1(-l)"akep~)(z, 0). k-O

or in other words

K[fez)] == f(- DJ [ep (z,

u)] 1".0'

(9.28)

where Du is the operator of partial differentiation with respect to u. Conversely, to every entire function of two variables tp(z, u) there corresponds an operator defined by equation (9.28) which is continuous on a certain set of entire functions. We will impose on the operator a restriction somewhat weaker than continuity. An additive homogeneous operator, defined on some class of entire functions V, will be called regular if on each of the functions f(z) of the class V the sequence {KCI,,(z)]} converges uniformly in each finite region to the function Kl/(z)]. THEOREM 17. In order that the operator K(f(z)] be a regular it is necessary and sufficient that it be representable in the form

~·-operator.

(9.29) K (f(z)] == f(- DJ [ep (z, u)] 1... 0' where Cf(Z, u) is an entire function of two variables of the class PI, admitting the representation (9.30) in ....hich Yl ;> 0, and w(z, u) is a function of genus one in each of the variables ~er the condition that the second variable is fixed in the lower half-plane ('.e., in the representation (9.2S) )'. - 0).

374

OPERATORS PRESERVING INEQUALITIES

[CHAP. IX

PROOF. First we shall prove the necessity. Suppose that K is a regular ~~ operator. Applying it to the function r which for 1m u <; 0 belongs to p~. we obtain the function

U ,

!f (z, u) = K (e -lIB) EP:.

If K were a continuous ~:-operator, then by what has been proved above rp(z, u) would be a function in P~. We shall show that this is the situation only in the case of regularity of the operator K. From the regularity of K it follows that for any fixed u the sequence of functions
=
rJ

±(l-~)"'k~l-~)
(
t

converges uniformly to q{z, u). Evidently if 1m z ~ 0 the function rp,,(z, u) is an H-polynomial in u. Without loss of generality we may suppose that qJo(z) is not identically zero. By the remark to Theorem 3 of Chapter VIII the polynomials qJ,,(z, u) form a normal family. It follows that for any fixed Z (1m Z ..;;; 0) this sequence converges uniformly, so that qJ(z, u) E for 1m Z ..;;; O. Finally, e-'" is, for 1m u ~ 0, a P~-majorant of the function rUz, and since subordination is preserved after application of a ~·-operator, therefore q(z, u) is a majorant of rp(z, u), and thus for 1m z ~ 0, 1m u ~ 0

P:

If(z, u)I>I
u)l·

Thus, cr(z, u) E P!, and by Theorem 14 it may be represented in the form

__

~(z)=;P(O, -z)= ~ 'fk~O)

(_Z)k,

(9.31 )

o

where rp,,(z) = !p(k){Z, u) 1.. =0 = K[z''](-I)''. Evidently 'I-'{z) EP~. Applying the ~·-operator K to the H-polynomial tP7I().:) for A > 0, we obtain 00

_

K(~n(AZ»)= ~(l-~)

... (1- k n 1) 'P1:(O~ik(Z) Ak,

1:=0

or putting

K

Z

= 0 we have ,.

(~n (b») 1.=0 = ~(1 k-O

-,}) ... (1- k n 1) I 'PI: ~OHI Ak.

(9.32)

Sec. 6]

!B

GENERAL FORM OF THE OPERA TORS

AND

!B *

375

In view of the regularity of the operator this sequence must converge for n -+ to the quantity

00

K (1jI (AZ)] 1.=0' We shall show that this is possible only if Y2 = O. Suppose first that rp(O, u) is not identically equal to zero. From the connection between the growth of an entire function and the rate of decrease of the coefficients of its power series (Chapter I, § 2) we have

and accordingly

lim }r 1'fldO) 11I).k = k~co V kl For any Y2

2"(II A•

> 0 we may choose A. > 0 such that 2y2A. > lim K (~n (Az)ll.=o =

1, and then we obtain

00.

n~oo

Thus, in this case it is proved that Y2 = O. If rp(O, u) == 0, then rp(z, u) = Z"rpl(Z, u), while rpl(O, u) '¥:- 0. :n this case one must take as tp(z) the function IjI (z) =

~l (0,

-

z)

= ~I ~~) (v,

- z) Iv=o'

Then we will have tp" (z) =
1,,=0 = zPO" (z)

(k = 1,2, ... ), while not all the 0k(O) are equal to zero. Substituting .p,,(,l,z) into the operator, we obtain n

___

KI~n(AZ)]=~(l-~) ... ( l - k n

l)zp

8k(O~~k(Z) A".

k=O

From the regularity of the operator it follows that the sequence

{z- pK (~n (AZ; 1.=0' has a finite limit. However. this is possible ')nly in case Y2 = O. The first part of the theorem is proved. Now we shall prove that if q;(z, u) is any entire function of the form (9.30), then the operator defined by the equation (9.29) is a regular ~*-operator. SUppose that f(z) E p* and ]n(z) is the result oi the application to f(z) of a Jensen operator. ],,(z) is an H-polynomial and therefore defines a continuous !8*-operator ],,( - D,,). Defining

c\I,. (z,

u) =

in (- D.) (cp (z,

u)).

376

OPERATORS PRESERVING INEQUALITIES

we obtain, from Theorem 15, that lp,,(z, u) in powers of z, we will have

,(z,

D)

E

P;.

[CHAP. IX

Expanding q;(z, u) into series

== 1)0 (D) + '1:~) Z + ~ zt + ... ,

where (}o(u), (}l(U), ... are entire functions of class p. and of genus no greater than one. Applying the operator j,,( - D.J, we obtain

'l'tI (z,

-

D) == Itl (- D) [&0 (u»)

+ "ifz I,.- (- D) [&1 (u») 81-

+~/tI(-D)[&I(D»)+ .. ' .

•

From the simple estimates (9.16) and (9.17), obtained in the proof of Theorem 8, it follows that the functionsj.. ( - D)[{}k(u)] converge uniformly, as n -+ 00, to the functions f( - D)[{}k(U)]. If for a given u(lm u <; 0) not all of these functions are equal to zero, then by the remark to Theorem 3 ofC!,apter VIII the sequence {lp,,(z, u)} of functions of class P~ forms a normal family, so that they uniformly converge to functions 00

1(-DJ[,(z, u»)=

~/(-D")[&II(D) ~:],

o belonging to P~ for u fixed and in the closed lower half-plane. Putting u = 0, we will have for fez) E p.: K [/(z») = 1(- D,,) [
(9.33)

while

K [/ (z») = 11m [Yn (z, ,.-+00

u»)I","o =

11m K

n-+oo

[i,. (z») ,

i.e., K[f(z)] is a regular !a·-operator. The theorem is proved. We turn to the general form of a !a-operator. We obtain it also under a certain additional requirement related to the continuity of the operator. We will say that the sequence {p"(z)} of functions of exponential type converge with the weight A (A > 0) to the function of exponential type ..t . Po(z)(P,,(z) - - Po(z», If

11m (sup I P,.(z)-Po(z)1 r A1 • 1)== O.

,.-+ ex>

(9.34)

•

The operator K over functions of exponential type will be said to be upper

continuous, if to each positive number A there corresponds a positive number B such that from A

P,. (z)-Po(z) it fonows that

K [P,. (z)] -

B

K [Po (z»).

GENERAL FORM OF THE OPERATORS

Sec. 6]

THEOREM 18.

~

AND

~.

377

An upper continuous16 ~-opera'or X[f(z)] has the form

K [fez») = f(- DJ [, (z. u») 1".0.

(9.35)

where the entire function cp(z, u) of two t'ariables satisfies the conditions: (I) it belongs to the class Pl offunctions of two l'ariables; (2) it is of exponential type in z for bounded u, i.e., Urn l·j ... <XI

Inl~~iU)I=aB
(lul
(9.36)

Conversely, if the function cp(z, u) satisfies these conditions, then the operator Kif(z)], defined by formula (9.35), is an upper continuous ~-operator. Thefunction !p(z, u) is defined here by formula (9.26). PROOF. First of all we note that a continuous ~-operator Xif(z)] may be extended by continuity to the linear envelope of the class p.. Indeed, approximating the function F(z) of the class p. by a sequence of H-polynomials !,,(z), we obtain a uniformly convergent sequence {X[!,,(z)]} of entire functions of class p.. Extending the operator by continuity, we obtain a ~·-operator. From'Theorem 17 this operator has the form (9.35), while ,(z. u) = K [e-'"]

and !p(z, u) E Pl. It remains to prove that the function cp(z, u) satisfies condition (2). To this end we use the upper continuity of the operator X. The set {e- W } of functions of z is bounded in the norm

IIfilB = for

lui <; R.

sup If(z) I,-BI.j

•

We shall show that for some positive B, depending on R, and

lui <; R, sup I ,(z. u) I e- B I • I ~ M.

•

Indeed, if this were not the case, then for any M be a point u'(lu'l < R) at which sup !'P(z, u')!e- Bj ' l

•

Choose a sequence B" SUch that

00,

M" -

supl,(z,

But for

-

•

Un -

00

(9.31)

> 0 and B > 0 there would

> M > O•

and a convergent sequence of points u"

uJI,-Bllj·I>M".

(9.38)

110 (11101 <; R)

sup! ,-"n' _,-IIe"

•

,-AI'I~

0

(A~ R) .

, 1: It is not dear how essential the requirement of upper continuity is in this theorem, just II Yrith the requirement of regubirity in the preceding theorem.

378

OPERATORS PRESERVING INEQUALITIES

Accordingly, for some

[CHAP. IX

B> 0

suplcp(z. Un)-cp(z. uo)Ie-B'.I~O

• > 0, beginning with some n, sup Icp(z. un)1 e- B I zl 0).

which contradicts (9.38). Thus, (9.37) is proved, which evidently is eq'uivalent to condition (2). To prove the second part of the theorem we suppose that q;(z, u) satisfies conditions (1) and (2), and we define an operator over functions /(z) of class P in the following way:

K[f(z») =/(-Du)[cp(z. u»)lu=o

= 2~l J.HU) cp (z.

-

u) duo

(9.39)

(OR)

where tp(u) is the Borel transform of the function /(z), R a number higher than the degree of the function/(z), and ell a circumference with center at the origin and radius R. By Theorem 17 this operator is a
where Nt is a constant depending on the function/(z). Accordingly, the operator K[/(z)] carries entire functions of exponential type into functions of the same class, i.e., it is a
Sec. 6] and

GENERAL FORM OF THE OPERATORS ~ AND

!B*

'P (Z, u) = e-'" ~ (u).

379

(9.41)

By Theorem 17 (u) is a function of class P* and of genus one. It is easy to see from (9.41) that on functions of the form e-tu the operator K coincides with the operator <1>( - D). The operators K and 4>( - D) coincide also on all polynomials and hence on the entire domain of definition. One obtains a theorem: 19. A regular 1J3*-operator, commuting with the differentiation, operator, has the form !B. [/(z)] = F (D) [1
where F(z) is a function of class p* and ofgenus one.

Analogously one obtains the following theorem. THEOREM 19'. An upper continuous lJ3-operator, commuting with the differentiation operator, has the form

!B [f (z)] = F (D) [f (z»), where F(z) is any function of class P*.

(2) We consider further operators K[J(z)] commuting with the operator z(dj dz). Putting we will have

Zepk (z) = K [kzk]

=

kepk (z),

or 'Pk (z) = "(k Zk •

Applying the operator K[f(z)] to the polynomial

We obtain

"

K[Q(z») = ~

"(takzk.

k-O

. IS

If K is a ~- or a ~*-operator, and Q(z) an H-polynomial, then K[Q(z)] a~so an H-polynomial. Hence it easily follows that the operator K[f(z)]

carnes every polynomial with positive roots into a polynomial with positive rOots.16 Accordingly, 1'0,1'1,1'2' ... is a sequence of multipliers of genus one and all of the same sign. Thus one obtains ~~_In order to prove this, it suflic:es to consider the polynomial K(p(e'Yz)] (-7r --.c P(z) is a polynomial with positive roots.

< " < 0).

380

OPERATORS PRESERVING INEQUALITIES

[CHAP. IX

THEOREM 20. Et'ery ~- or ~·-operator, commuting with the operator z(d/dz), coincides with the operator r defined in Chapter vm.

Knowledge of the general form of ~- and ~·-operators makes it possible to construct an infinite set of such operators. In particular, if w(z) E!', then the operator K[f(z)], defined by the equation

K (e-IU) = 01 (az + bu) is a

~-operator,

(a

> 0,

b> 0).

satisfying the condition a~

[zl(z») = b {~[/(z)J}'.

7. Certain Extremal Properties of Entire Functions In this section we shall prove some theorems related to entire functions of exponential type, least deviating from zero, which are generalizations of corresponding theorems of S. N. BernsteIn,1' Suppos~ that w(z) is a function of class' and exponential type CI",. For functions f{x), defined on the real axis, we define the deviation from zero with weight Iw(x)1 as

-

Bill [ I ] THEOREM

or equal to

CI...

I/(x) I sup I• -co
(9.42)

21. Of all the real entire functions of exponential type less than and of the form

l(x)="o+a1x+ ... +an_lxn-l+xn+att+lxn+l+ ... (9.43)

>

(n 0). the unique function giving the smallest deviation from zero with the weight Iw(x) I is the function ID(X)

10 (x) = nl Re lD(n) (0) PROOF.

•

(9.44)

For the functionfo(x) we have

Ew [/0] = , Wl,.~t(O) " Suppose that some entire function f(x) of form (9.43) of exponential type not larger than CI. satisfies lIt

I/(x) 1<, ID(") (0)1 I 01 (x) I

(- 00 < x < 00).

Then the function OIl

(z)

=

lit (0)

11)(")

01 (z)

17 See S. N. Bemlteln [II. Chapter Ill, Theorems I and II of § I and Tbcorems II and III oft 9.

CERTAIN EXTREMAL PROPERTIES OF ENTIRE FUNCTIONS

Sec. 7]

381

is a majorant for f(x), and since

I(n) (0) = therefore by Theorem II f(x) REMARK.

w~n)

(0).

= Re w1(x) == Io(x).

If one assumes n

== 0, i.e., fixes

the free term in the expansion

I(x) = 1+a 1x+a2x'+ ...• then evidently the function • (z)

10 (x) = Re • (0) gives the smallest deviation L == I/lw(O)1 to the expression (9.42). However, in this case one can not assert the uniqueness of the extremal function.!8 THEOREM 22. Suppose that w(z) = s(z) + it(z) is a function of class P and of exponential type (1 ;> 0 without real roots, not equal to the exponential function ce±az or to a polynomial of degree less than or equal to n. Then among all the real functions of exponential type T <; (1 of the form

!(z)=aO +a 1 z+ ... +an_ 1z"-1+az"+bz"+1 +an+~+lI+ ... (n~ I).

(9.45)

where a and b are fixed, the unique function giving the smallest deviation E ...[J] is the function (9.46) 11 (x) = As (x) Bt (x).

+

where A and B are chosen so that h(x) has the form (9.45).

PROOf. First of all we shall show that one may choose constants A and B such that the functionh(x) will have the form (9.45). This problem reduces to the problem of the solvability of a system of equations Asln) (0) Bt
+

+

+

i.e., to the proof that the determinant A = sln+1) (0) t
is not zero. By the hypothesis of the theorem, w(z) has no real roots. From this hypothesis it follows that W(II'(Z) also has no real roots (see (9.11», and accordingly W(II)(O) ~ O. We have the equality 1m .(n+ 1) (0) .(n, (0) -

b.

I .(n) (0)11 •

'8 For instance. for w(z) = e i • all fUl1Ctions/(x) = cos Ax (IAI to /(0) = I.

~

I) give a deviation equal

382

OPERATORS PRESERVING INEQUALITIES

[CHAP. IX

If 1m [W(II H)(O)/w(II)(O)] = 0, then, as one easily sees from the expansion (9.11) of the logarithmic derivative of the function w(")(z) of class P, W(II)(Z) is a real function. Since it can not have any real roots, it therefore has no roots at all, so that .J'" (z) ce'el •

=

The equation k = 0 is impossible, as in this case w(z) = P,,(z). For k ¢ 0 Q)

(z)

=

k~ eb

+P

"-1 (z).

(9.48)

The case P n-I(Z) == 0 is excluded by the condition of the theorem. A function w(z), represented by equation (9.48), grows on the imaginary axis like a polynomial. Therefore it is of completely regular growth. Its indicator diagram is a segment [0, k] of the real axis, and accordingly it has roots in both halfplanes. This contradicts the statement that w(z) E P. Thus, .(,,+1) (0)

1m

("I

II)

(0)

+0

so that the system (9.47) has a unique solution. Now suppose that a function of type (9.45) satisfies on the entire axis the inequality

I~~;~ I.(:E

w

(/1)'

whereh(x) is defined by equation (9.46). The function

X(z) = fez) - (A -iB) Q) (z) belongs, by Lemma I, to the class P, while

& ·(x)

= RX(x) = Co + c1x+ ..• + C"_lx"-l+ C,,+~"H+ ..•.

If D(z) ~ 0, then all the roots of D(z) are real. But then all the roots of the Jensen polynomial Dn +2(z) are also real. On the other hand, by the Laguerre theorem, a polynomial for which two successive coefficients are zero (other than the first two or the last two) must have nonreal roots,I9 and therefore either c"+11 = 0 (q = 2,3, ... ), or Co = CI = ... = Cn - I = O. The first case is impossible, because then

I [(A - iB) Q) (x) )

= I [X (x) ) = P ,,+1 (x)

and the function w(x) is a polynomial of degree not larger than n + 1. If now Co = CI = ... = Cn - I = 0, then the function D(x) has at the origin a root of multiplicity not less than three. From the interlacing of the roots of the real and imaginary parts it follows that (A - iB)w(x) has a multiple root at the origin. This is impossible, since w(x) has no real roots. Thus, f(x} = Re l(A -IB)Q)(x»). The theorem is proved. II

Chapter VIII. Lemma 3.

ApPENDIX

I

SOME ADDITIONAL PROBLEMS OF THE GENERAL THEORY 1. The Impossibility of CoDStruetiag an Exact Scale of Growth The scale of growth (of order and type) usually used in the theory of entire functions is very incomplete. Functions with the rates of growth, for example, of e r and rne r have the same order and type. The rates of growth of functions of infinite order can differ even more widely. The question naturally arises whether it is possible to construct an exact growth scale, so that to every monotonically increasing function there will correspond a positive number in such a way that a function with a greater rate of growth will correspond to a larger number. Here we assume that the rate of growth of
The impossibility of finding such a scale is a consequence of the following theorem of Du Bois-Reymond. THEOREM.

Let '~l (r), 1j19 (r),

... ,

~n (r),

., .

be a sequence of positive, monotonically increasing functions such that the rate of growth of any function is greater than that of the preceding function. There exists a positive, monotonically increasing function 1p(r) with a rate of growth greater than that of any member of this sequence.

In fact, the convergent series (X)

~ (r) = ~ 2no/~ (II) ;" (r) "~1

defines a monotonic function, and for any n we have '" (r)

"'n (r)

>

1. 4"+1 (r) _ 2"+1 o/n+l (11+1) "'" (r)

00

for

T _

00.

~he introduction of transfinite numbers does not help, since if we consider the .Inverse functions, we can state that, for any sequence of positive, monotonically increasing functions in which the rate of growth of any function is 383

384

SOME ADDITIONAL PROBLEMS OF THE GENERAL THEORY

[APPENDIX I

smaller than that of the preceding function, there exists a positive, monotonically increasing function of smaller rate of growth than that of any function of the sequence. 1 1. COIlYerpllt aad Dl'eI'Iellt Typesl To give a more precise characterization of the rate of growth of functions of finite order, we introduce the idea of convergent and divergent types. An entire functionj(z) of order p is said to be of the first or second type, depending on the convergence of the integral

f

00

in Mf(r)

,.,+1

(1)

dr.

".

If p

is not an integer, then the convergence of the integral (I) is equivalent to the convergence of the series THEOREM.

00

~ la~IP'

(2)

le-l

where al: (k = 1,2, ... ) are the zeros of /(z), excluding a possible zero at the origin. If the integral (I) converges, then from (1.27) the integral

(3) also converges. We also have

J ,. co

II

(r) ./ p ~

r'

(t) dt 1'+1'

II

and so, since (3) converges,

11m

II

,.+00

Thus

~ -L_

f

i-I

0

~

,. I' fJ

00

(r) _

_"(tl!+pj CD

411(1)

I'

o.

,. CD

I'

00

1(t)4r==pj"(t) dt. tf+l

1'+1

(4)

I We note that it is also impolllible to constnlC:t an exact growth scale for the subclass of the monotonic functions consisting of those func:tions that are the maximum absolute values of entire func:tioDs. To prove this, with each monotoDic: function tK,z), tendiDs to iDfiDity, we

usoc:iate the function f() z -

~.fell ~

II-I

where the numbers p. are chosen so that 1im[¥<11 00.

• See O. Valiron [I).

+

,.,.

I) ,.

z ,

+ IW'" - I. It is clear that lim M/.')/¥<') -

Sec. 2]

CONVERGENT AND DIVERGENT TYPES

385

and the convergence of the series (2) follows from the convergence of the integral (1). Conversely, (4) shows that the integral (3) converges when the series (2) converges. Further, it follows from Hadamard's theorem and (1.24) that InM,(r)
(f r

II (I)

,+1

JCD II (I) ) dt+r r ,+I dt +0(rJ').

Dividing both sides of this relation by r,+I, integrating from some lower limit > 0 to 00, and changing the order of integration in the right hand side, we obtain CD InMf(r) r P+1 dr 00 • at

f

<

•

We note that, when the order p is an integer, the preceding reasoning remains ·valid except for the fact that (1.40) and (1.41) must be used instead of (1.24), and we then obtain the result: In the case of an integral order p, the convergence of the integral (1) is equivalent to the convergence of the series (2) and the integral (5)

It is not difficult to generalize this theorem to the proximate order p(r), by replacing the integral (I) and the series (2) by the integrals

f

co

rP1r)+1

J co

In M,(r) dr,

dn (I) tp I"

,

and the integral (5) by co

J18jr)1

-(-) dr.

rL r

The theorem we have obtained can be used to give a very simple proof of the following theorem due to S. N. Berniteln [6]. THEOREM.

A. sum of even entire functions of genus zero is an entire function

of genus zero. Let f(z) and q(z) be even entire functions of genus zero. Then F(z) = f(Vi) and cJ)(z) = q(Vi) are entire functions of order not higher than half that of the original functions, and their zeros {b.} and {b~} are the squares of the zeros ofj(z) and q(z). PROOF.

386

SOME ADDITIONAL PROBLEMS OF THE GENERAL THEORY

[APPENDIX I

Since I(z) and tp(z) are of genus zero, the series

converge, and thus F(z) and ct>(z) are of convergent type with p = But if , is large enough, M~') > I, M /b(') > I, and Mp+. (r)

It immediately follows Therefore the series

in which the

Qn

<: MF (r) + M. (r) ~ 2Mp (r) M. (r). that F(z) + ct>(z) is of convergent type

1.

with p

== 1.

~-11' lanlz

are the zeros of F(z)

+ ct>(z), converges, co

F (z)

+ ~ (z) = cz'" II (I A= 1

and the entire function f(z)

: ) /I

co

+
CZ2",

II (1 _ :2) A=1

n

is of genus zero. We note that the requirement thatf(z) and tp(z) be even is essential; there are examples of entire functions having the same genus, the sum of which has a genus greater by one than the genus of the original functions. Such an example was given by Lindelof in [3].

3. De Paley-Wiener 1beorem In Fourier-transform theory there is a well-known theorem due to Plancherel :3 Ifl(x) E L2( - 00 < x < (0), then 1 'P(A)=I.i.m' ..r=

r

'It

f"

f(x)e i 1.rdx'

-II

• See, for example, Titchmarsh [1]. t The relation !p(x) = lim 9'.(x) denotes that

f""I,

11m .... co

(x) - , . (x) Ildx =0•

-uo

In the present case, the convergent sequence is the sequence of functions 2'. '2'2' (i.) J

II

= ..rJ-::K

j/(X)ei'A:ll dX -2',

(T1 ....

+oo aDd T1 .... +oo).

(I)

Sec. 3]

387

THE PALEY-WIENER THEOREM

exists, and

I (x) =

.

fn . ',-

1

I

(2)

I.t.m. ,rqJ (A.) e- t ..... d", r 1t -n

almost everywhere. Moreover co

co

f I/(x)l~dx= f IqI(A)I:JdA. -co

(3)

-co

We will use this theorem in the proof of the following theorem due to Paley and Wiener [1].6 THEOREM.

For a function f(x)( -

00

< x < (0) to be expressible in the form

/)

I(x)

=

f ei)..l'~ (A)

(~()..) EL:J (a, b).

d)"

(4)

a

it is necessary and sufficient that fix) be square-integrable along the whole real

axis, and that it be possible to extend the definition off(x) to the complex plane as an entire function of exponential type. If the interval (a, b) can not be replaced by a smaller interval, then the segment (ia, ib) of the imaginary axis coincides with the cOnjugate diagram off(z). PROOF.

Let

II

I(z) =

f

ei ).z1' ()..) dA.

(4')

a

where tp(A) E L2(a, b). The function tp(A) can be approximated in mean square by continuous functions " ..(A). Thus

I

f(z) -

II

f ei).z~n()")dA I~ e

C

/)

Ilnu.1

a

f I'HA)- 9n (A) Id)" a

~ ecilmil

v'

(c=max

(h-a)

(j 19()")-~.(A)I'Ad)") a

(Ial. Ihl».

On the other hand, every integral of the form /)

I. (z) =

f eil.c 1\1. (A) dA a

cafi~ be uniformly approximated in any bounded region of the z plane by the nlte Sums _ ~ ei).~I\IJI (A~ (Ak+l - A~. • The following proof of this theoRlll was Biven by M. Plancherel and G. P61ya in (I).

388

SOME ADDITIONAL PROBLEMS OF THE GENERAL THEORY

[APPENDIX I

Thus fez) is the limit of a uniformly convergent sequence of entire functions and is thus an entire function. Moreover, it follows from (4') that b

f

If(z) 1-<. e-~ 1m. I '\I (A) IdA -<. Me-a 1m

I'

a

for 1m z

> 0, while for 1m z < 0 we have If(z)I-< Me-" 1m,

(M

= f"1'\1 (A) Id A) • a

It follows from these two equalities that the indicator diagram of f(z) is a segment of the imaginary axis contained in the segment (-;b, ;a). Now let fez) be an entire function of exponential type and let f(x) E L2( - 00, 00). We write co

,(z) =

It follows from (3) that

f e-u:f(x)dx o

(Re 2'

> 0).

co

f I,

(II + It)- ,(Ig+ It) Ii dt

-co

For 8 1 - 0 and E2 - 0, the integral on the right tends to zero and thus, for 8 - +0, the function !P(E + it) tends to some function 'P+(I) E L2( - 00, 00) in the mean square sense. If we define the function !p(z) for Re z < () by CD

,(z)==-

f ,af(-x)dx, o

we find, as above, that for 8 < 0 and 8 - 0 the function !P(8 + il) tends to some function 'P_(I) E L2( - 00, 00) in the mean square sense. The function !p(z) obviously coincides with the Borel transform of f(z) (Chapter I, § 20). Taking for the path of integration a rectangk containing in its interior the conjugate diagram (ia., ;(J) of the functionf(z), from (1.120) we obtain

fez)

= 2!1 J'e(H")I"(I+lt)dt-~1 j'e<-HU),,(_a+ It)dt ~

+

~i

•

-.f

e
~

f-.• 1', (s+I~I)

~l

e(I+",)

ds

Sec. 3]

389

THE PALEY-WIENER THEOREM

Since r{z) is holomorphic everywhere outside the conjugate diagram, for 0 the last two integrals here tend to zero, and p,

E -+

2~

f (z) =

f e'b (,~+ (t) -

~ _ (t)J dt.

'" For points of the imaginary axis outside the conjugate diagram, the limit values 'P+(t) and 'P_(t) are equal, and thus p

f (z) =

2~

f

eilll

~ (t) dt

(~(t) = ~+ (t) - ,~_ (t».

" The theorem has thus been proved. The Paley-Wiener theorem has the following generalization. THEOREM. Let the indicator diagram I of the entire function f(z) of exponential type be polygonal. Let the rays arg z = OJ (j = 1,2, ... , n) be parallel to the normals of the sides of this polygon, and let the functions

i'.

fJ(x) = f(xe .1) e

-la(' IQl

J

belong to the class L:I(O, 00). Then

f

eU ~ (A) dA, (5) r where r is the boundary of the conjugate diagram, and ?p(A.) is a holomorphic function outside 1 such that

f(z) =

fl~(A)I')ld"'
(6)

r

With no loss of generality, we can assume that 8; 0. Then the function

PROOF.

h(Oj)

=

=0

and

co

CPj(B+it)=

f r(Hit)a:fJ(x)dx

(8)0)

o belongs to L:I(-oo, 00) and tends to some function 1p/(t) E L:I(-oo, 00) in the mean square sense when E - +0. The functionf(z) can be expressed as an integral taken around the perimeter of a polygon 1. that contains the conjugate diagram of the function in its interior (Figure 9): f(Z)=i..c

r

(A~DA)

,ucp(A)dA.

By Cauchy's theorem, we can replace this integral by the integral around the COntour KHGCDEK. .

390

SOME ADDITIONAL PROBLEMS OF THE GENERAL THEORY

[APPENDIX I

Bya passage to the limit, as in the proof of the previous theorem, the integration along KH can be replaced by an integration along EG. It can similarly be shown that, for E sufficiently small, the difference between the integral along GCM and that along LM can be made as small as we please, C r-_

Mr-

,, I

--_c II

Fig. 9

and, since E > 0 is arbitrary, the integral along FGCM is equal to that along FM. By a continuation of this process we find that the integral along ABCDA is equal to that about 1. Moreover, the limiting values of q:(z) on each of the sides are functions of L2. The theorem is thus proved. s It follows from the above theorem that every function of exponential type satisfying the condition of the theorem can be expressed in the form ~ l..

il.

f(z):== ~ e 3 gJ (-Ie Jz),

i=l where the Ai are the vertices of the conjugate diagram, and the entire functions gtz) are of the class L2( - 00, 00) on the real axis. In fact, the integral along any side L j of the polygon Ai + te i6, (0 < t OS;;;; d,) can be written fi (z) =

or

fe

'~i ().) dA.

).z

•

'r"

= eL.3 e'9J

Lj

. exp (te i'3Z) '~i ().i

+ te J) dt i9

0

where gi(U)=ei'j

" leitd~i()'+tei'j)dt, o

• We note that the converse is also true, i.e., if an entire function of exponential type is expressed by the integral (S) and tp(A) satisfies (6), then all the f,{x) (j = I, 2, ... , n) belong to L ,(0, 00).

Sec. 3)

391

THE PALEY·WIENER THEOREM

i.e., g/u) is an entire function of exponential type, belonging to L2( - 00, 00) on the real axis. Liht [1] extended the theorem proved here to the case when the indicator diagram is a disk. He showed that in this case a necessary and sufficient condition for an entire function of exponential type to be represented in the form (5) is that

102ft {Xl Yx e-

2II(9)xlJ(xe itl W

dx dfJ

< 00.

It is clear from this result that the hypotheses of the Paley-Wiener theorem must be modified if it is to be generalized to arbitrary indicator diagrams. Kacnel'son [1] gave a further generalization of the theorem when the indicator diagram of the entire function of exponential type is the disk Izl ~ R. He introduced the weight function

1

e-2t,

00

p(r)

=

r

--;:===~ dt

o Y2t + t 2

and proved that representability ofJ(z) in the form (5) is equivalent to

102ft1ooolJ(rei9We-2,Rp(rR)dr dfJ < 00, and this in turn is equivalent to having the Borel transform 1/1(1\) of J(z) belong to the Smirnov class E2 in the domain 11\1 > R. Finally, with this weight function there is an analog of Parseval's equation,

i

IAI-R

1~(1\W d1\ = i2ft 0

L

oo IJ(re itl We- 2rR p(rR)dr dfJ.

0

In the general case of an arbitrary indicator diagram, Kacnel'son showed that a necessary condition for the Borel transform ~(1\) of f(z) to belong to the Smimov class E2 in the exterior of the conjugate indicator diagram is that

102ft 10

00

IJ(reillWe -211(9), dr dfJ <; c, IrI~(1\Wld1\1

for some constant c,. If the indicator diagram is a polygon, this condition is both necessary and sufficient. To study the representability of an entire function of exponential type when the indicator diagram is arbitrary, Kacnel'son [1] introduced the weight function PrU') = I' - d)(nl + 1,- d2(nl in the exterior t of the conjugate diagram, where aM) and a2(n are the POints of contact of the Conjugate diagram with the supporting lines drawn

392

SOME ADDITIONAL PROBLEMS OF THE GENERAL THEORY [APPENDIX I

from the point rEt. For functions holomorphic in t he defined a scalar product using integration over t with weight Pr
(j, g) =

fo

2fr

fooo e- 2h(')'j(re il ) g(re il ) dr dB

and introduced the Hilbert space Hh(,). Kacnel'son showed that the operator

1 . fl/l(~)eAz f(z) = -2 c

'lTi

d~ III/Ilir.

(C surrounds f) maps Hr onto Hh(')' and IIfllh(') = 4. LeYitaD Polynomials

A very important theorem concerning the approximation by trigonometric polynomials of entire functions of exponential type bounded on the real axis was proved by B. M. Levitan. There are now several proofs of this theorem. 7 The proof we give is due to N. I. Ahiezer and V. A. Ma~nko [I]. THEOREM.

Let f(z) be an entire function of exponential type a and let

1/(x)I<:M For any N

such that

(I)

• k

~

I-II)

ak. "e "

.6.J

-(~H) < !. ,. < ~+.

< 'Ij

I/(x)-P,.(x) I

Ixl <; N, and I P.,. (x) I ~ M for PROOF.

< x < 00).

> 0, 'Ij > 0 and e > 0, there is a trigonometric polynomial P ,,(x) =

for

(-00

00

(2) (3)

< x < 00.

We consider the function If (x) = /(x)

(SI:;xr

(x> 0),

(4)

< x < 00),

(5)

which satisfies the inequality M

IIf (x) I <: ~3x3

(-00

and construct the series 00

1\17'(x)

=

~ 'P(x+nn,

(6)

,,=--~

which converges absolutely on the real axis. The number K > 0 can be chosen so small that

If(x)-~(x)1

<.;.

'See B. M. Levitan [zl. M. G. KreIn (5).

for

Ixl <; N.

(7)

Sec. 5)

393

POWER SERIES

If T is chosen so that KT is a multiple of 11', we have co

~

, () _ sln'u IjIT Z -

f(z+ liT) (z+ IIT)2

~

.,.2

•

"~-CIO

It is obvious that

co

I ~T(X) I <. M sin2 xX ~

(u

"=--00

~ 11K)" = M.

To prove this theorem, we need only replace the factor' (sin KX/KX)2 by (sin KX/KX)2+1.:.

5. Power Series with Coe8idents that are the Values of EatIre FaactiOllS of Exponential Type for Integral Values Df Their Arguments We consider series of the form

co

f(z)=~,(n)z",

,,=0

(1)

where rp(z) is an entire function of exponential type. The distribution of the singularities of a functionJtz) expressed in this way is given by the following theorem of G. P6lya. THEOREM. If the width of the indicator diagram h.,(7J'/2) + h.,( -11'/2) of qJ{z) is less than 27J', then all the singularities ofJtz) are in a region G obtained from the cOnjugate diagram I of rp(z) by the transformation

andf(oo) = o. Conversely, series

'lII=e- 1I

(2)

ifJtz) can be expressed in the neighborh9Od of the origin by the co

fez) = ~ a"z",

.=0

(3)

all its singularities are in some region G obtained by the transformation (2) of a convex region K with width along the imaginary axis less that 211', and ifJt 00) == 0, then there exists an entire function cp(z) of exponential type with its conjugate diagram in K such that cp(n) == a. (n == 0, 1, 2, ...). For the proof of the first part of the theorem, we note that an entire function !p{z) of exponential type can be written in the form

cp (u) =

Je + ul

(A) dA.

((11

Where C is an arbitrary contour encircling the diagram 1. We may, of course, assume that C is the boundary of a convex region with width on the imaginary axis smaller than 211'. • By introducing a new kernel to replace (sin a/a)l, we can generalize Levitan', theorem 1bia was done by V. A. Mari\cnko

!'> the case oft\mctiona with rapid growth on tile real uia.

See also the artic:Ie [I] by L I. llookiD. I,jubuskii [I] bu recently obtained a more paeraI reauIt on the approlIimation of entire functioDa of exponential type. in various norms, by trigonometric polyDomiaJs.

In [%].

394

SOME ADDIIIONAL PROBLEMS OF THE GENERAL THEORY

[APPFNDlX I

Substituting this expression for 9'
t(z)=

f ~(A)(~"so,ftl.z")dA.

«(1)

In a sufficiently small circle and so

Izl

< E,

fez) =

the series in the integrand is convergent,

f l-ze dA. (a, 4(1..)

l

It follows from this representation of fez) that it has no singularities outside the one-sheeted region obtained by the transformation (2) from the region bounded by the contour C. In other words, all the singularities of fez) are in the image ofG. From this representation off(z) it also follows thatf(oo) = O. Conversely, let all the singularities of a function fez) given in the form (1) lie in some closed region G, where this region is the image of a convex region K with width less than 27T in the direction of the imaginary axis. In this case, the set complementary to G contains the points and 00. We have

°

I an = 2nl

Jf(Z)

z,i+ 1 dz

(n

= 0,

1, 2, ... )

( I

(C is a circle of small radius with center the origin). Since we also have f( 00) = 0, by the fundamental theorem of residue theory we may replace this integral by an integral around any contour L that encircles the region G but does not encircle th~ origin. If

then

'f(U)=~lf f(e-

fJ

)e'4tJ dv,

(4)

«(1)

where C is a contour encircling the region K. It is evident that 9'
Sec. 6]

LOWER ESTIMATES FOR HOLOMORPHIC FUNCTIONS

394A

In the work of A. O. Gel'fond [3], a substantial improvement in Wiegert's theorem is obtained. A relation is established between the rate of growth of the function tp(z) and that of the entire function G(z), where these functions are connected by the relationj(z) = G(l/(z - I». 6. Lower Emmates for Holomorpbic Functions An estimate cruder than Cartan's (given in Chapter I) had already been established by Valiron [1]. Ahlfors [1] gave a similar lower bound for subharmonic functions in the plane with a finite total mass n. The difference between this inequality and Cartan's is that the sum of the radii of the excluded disks is 5H rather than 2H as in Cartan's inequality.lI Landkof [1] obtained a similar inequality for subharmonic functions in an arbitrary k-space, with the sum of the radii of the exceptional disks equal to CkH, where Ck depends only on k. Lower bounds for the moduli of holomorphic functions in a disk are given in Saginjan [1; 2], Havinson [1], and U§akova [1-3]. We also mention lower bounds in Izl ~ 1 for the modulus of a function that is holomorphic and bounded in Izl < 1 (Govorov [4], Grisin [3], Kudina [1]).

In the first group of results there may be infinitely many exceptional disks, all inside Izl < 1; in the second group, there are finitely many but they may extend outside the unit disk.

II In 1968 GriSin [1] obtained the same inequality outside exceptional disks the sum of whose radii is 2H. as in Cartan's inequality.

ApPENDIX

11

THE APPLICATION OF UNIQUENESS THEOREMS TO QUASIANALYTIC CLASSES OF FUNCTIONS The proofs of some uniqueness theorems in the real domain, for functions expressed by ordinary or generalized trigonometric series, can be obtained by reducing the problems to those of the uniqueness of the definition of holomorphic functions, i.e., by their reduction to problems of the type investigated in Chapter IV. In particular, some theorems concerning quasianalytic classes can be obtained in this way. 1. FunctioDS DefiDed by Their Values OD aD IDtenal

A periodic function is defined by its values on any interval of length equal to its period. But this property is not restricted to periodic functions. We can cite the examples of almost-periodic functions and functions represented by Fourier-Stieltjes integrals, which are also defined by their values on a sufficiently long interval. This property is related to the spectrum of the function. l We consider the class Tn ofStepanov almost-periodic functions (Sp),2 whose Fourier exponents make up some set n of real points. We denote by nn(t) the number of nonzero points of the set in the interval (-t, t), and write

f B

Ng(R)==

ng/') dl,

(1)

o • 1 M. G. Krein brought this relation between the spectrum of a function and its uniqueness mterval to my attention in 1941. 1be results given here were published by me in 1950 [11). New results in this direction were recently obtained by J. P. Kahane [1; 2J. These results are connected with the theory of functions "periodic in the mean." I For our purposes it is sufficient to use the following property of S.-almost-periodic functio?s, which can be taken as a definition. Every S.-almost-periodic function I(x) can be approximated by an almost-periodic polynomial P(x) in the sense that

z+1

sup -ao
If

1

I/(t)-P(t)I"dI 1

I

III

395

'
396

APPLICATION OF UNIQUENESS THEOREMS

when zero is not in the set

n, and

[APPENDIX II

j' -t-(I) dt+ In R, R

Ng(R)=

ng

u

when zero is in n. In particular, we can have No(R) = co. The function N o(R) characterizes the density of the distribution of points of n on the real axis. 1. !fan S'P1unction of the class To

THEOREM

00

~ akeiAkt

! (t) --

k=

(Ale EQ)

(2)

-00

is zero on some interval of length d and if

RI~moo [ Nil (R) - ~ R - ~ In R] = then f(x)

00

(~ + ~ =

1),

(3)

== 0 almost everywhere.

Thus a function of To is determined by its values on an interval of given length if the density of is sufficiently small.

n

PROOF. We obviously need only consider the case when n has a condensation point at infinity only. The Laplace transform of f(x) is 00

00

fe-tf!(t)dt= ~ ale. (4) ~ z-IAk o k=-oo and in this case is a meromorphic function with simple poles at the points O'k' We may assume, with no loss of generality, that f(l) is zero on the interval (-dj2, dj2).3 We first obtain an estimate of the size of tp(z) in the right halfplane. For m ~ 0 we have ((i(Z)

111+1

If

=

111+1

rtf! (t) dt

/-< f

til

111+1

e- tx I/(t) 1dt

-< 111/1" (f

1

e-qtz dt

}i,

,II

til

where z+1

II!L,= Hence

[f

sup

1

1!(t)I"dt]i" (Z=X+iy and ;

-00
If

rzt!(t) dt

I-< II!I~

+~= 1).

.!. e-

fIIZ

til

• This follows from the fact that the translation 1(1 the function.

(l-_:-qX)q (qx)q

+ '0> does not alter the spectrum of

Sec. I]

FUNCTIONS DEFINED BY VALUES ON AN INTERVAL

397

From this result we infer the convergence of the integral in (4) and obtain II

I

1

--3:

2

-

~l-e-q.1l)q

(qx)q(l-e-~)

for x

> O.

In the left half-plane we similarly obtain the result 1
-.!! 111' I 2

(! _e:-

1: ql 3: l )q

(5)

(q Ixl)Q (l-e- l 3: I)

which is valid for all values of z = x + iy. We note that iff(t) is bounded, in particular, if f(t) is an ordinary almost-periodic function, then this result can be replaced by (6)

where sup I/(t)I. 3a < t <00 Jensen's formula (Chapter I, § 5) for functions holomorphic in a closed circle Izl ~ R can be extended to apply to functions with poles in this circle. If the merom orphic function !p(z) is not zero either at infinity or at the origin, then Jensen's formula becomes M=

-00

R

B

2K

f

J

nt/t) dt n2/t) dt = 2~ In I

S

B

f

nil (t) dt

o

t

> !!..n R + !.q In R + 0 (1) +

B

f

ntCtl dt. t

(8)

0

3. The convergence of the series (4) has to be understood in a generalized sense. We know that there are mUltipliers

such that for a given e

{atr:."". 0 < at < 1 (8 > 0), > 0 and sufficiently small 8 > 0 we have

a:tl,

".

IIltt) It follows from (5) and (6) that

and "a --+

00

as /J --+ O.

~

-".

ata"ei.\o,'II, < e,

0 .. p ..

00.

398

APPlIC ATION OF UNIQUENESS THEOREMS

[APPENDIX II

(8') If zero is a point of the set Q and is also a simple pole of
: R -In R] = -

00

(3')

then f(t) == 0 almost everywhere. Theorem 1 has thus been extended to the case p = 00. 4 The estimate (3) is sharp, as is shown by the following example. For any ~ > 0, we write 11 (t) = (e"+ 1)-26, Ig(t) = (I r j ,)-26 for -'fr < 1 < 'fr, and h(t+21C)=/i(t) (1= I, 2; - 0 0
+

/(t)

= ei6'/1 (t) -

ria'/g(t)

belongs to the class (Sit) and is zero for It I < 'fr. The Fourier exponents of = ±(n + ~) (11 = 0, 1,2, ... ). We denote the set of numbers ±(n + ~) by Q, and obtain

f(t) are A. n

Ng(R) = 2R+(1-2Q)lnR+ 0(1).

Thus if we replace the quantity I/q in (3) by a larger number, we obtain a sequence Q satisfying condition (3), and an S,,-almost-periodic function f(l) not identically equal to zero. with Fourier exponents from Q, and equal to zero in the interval It I < 7T. It is of interest to investigate the "boundary functions," i.e .• the functions that are zero in the interval ItI < d/2 and such that their Fourier exponents satisfy lim S .. co

[M~(R)-: R-alnR]=-oo

(I+!~G>!). , For p

= I, instead of (5) we have 1,(z)I<;;U/U'

and in this case we must set 'I =

00

-!I arl I

(1_,-1 111 1)-1,

in the relation (3) of Theorem 1.

(9)

Sec. I]

FUNCTIONS DEFINED BY VALUES ON AN INTl:RVAL

399

In this case only one point need be omitted from n in order that there exist a function from Tn that is zero in the interval (-d, d) but not identically zerp. We have the following theorem. THEOREM 2. If the set {At} of Fourier exponents of an Sp-almost-periodic function f(t) satisfies condition (9), and if the function itself is zero on the interval (-dI2, dI2), then the set {At} is of density dl", on the positive and negative rays and

txists. Moreover the function f{t) can be expressed in the form co

let) -- ~ ¥ tn,.> ,o./cl

(10)

t

-GO

where I\'(z)=c

Um

B-+co

II

P,/cI
(1-

I: ).

(10')

/c

PROOF. It follows from the inequality (8) that n1(t) = 0 when condition (9) is satisfied. Thus q;(z) has no zeros, and so tp(z) = [q;(z):l1 is an entire function with zeros at the points of n. All the points of n must be zeros of tp(z), since it is impossible to remove any points from n. The representation

(4')

(see (4» shows, by a theorem of M. G. Krein (Chapter V, Theorem 12), that 1p(z) is of exponential type and of completely regular growth, and that its indicator diagram is a segment of the real axis. It thus follows that the set {At} has a density that is the same on the positive and negative half-axes, and that the limit

,\'_1

8= lim

.... A./c 1)./cI
B-+co

~xists. From Theorem II, Chapter V, the function tp{z) can be represented .l~ the form (10'). The representation (10) of f(t) is obtained from (4'), which Yields at = IJ(1p'(iAJ). We show further that the density of is d/2", on each

of the rays x

> 0, x < O.

n

It follows directly from (9) that lim B-+co

IIQ

(R)

R

d

~-=. ft

400

APPLICATION OF UNIQUENESS THEOREMS

[APPENDIX

II

On the other hand, (5) is a result of the fact thatf(x) vanishes on the interval (-dI2, dI2), and thus 1

IIJl(Z)I:;Pc

I Z I'(l-,-IQlll )

e{ III:!

( II)

(l-e- qIQl1 ),

By an application of Jensen's theorem, this result easily yields

. 11m

B-+co

ng (R) R

d

---:;p-. It

The set n therefore has the density dl1T, which means that its density is d121T on each of the rays x > 0 and x < o. The theorem is proved. The following theorem can be considered, in a sense, as the converse of Theorem 2. THEOREM 3.

fl

> Ofor k ¢

If ?p(z) has only the simple zeros iA/t n, and if, for y > I, tI

o < m < I~ (z) I e - 2" I

sl~ch

that inf IA/t - Ani

~

,

.l)

I (l

+ IziT) - 1 < M

(11 ')

outside non intersecting circles containing the zeros, then the series (12) -co

converges absolutely and uniformly on the whole real axis and defines a function f(t) which is zero on (-dI2, dI2).

PROOF.

It follows from (11') that

I Hz)

z-fAlc

1>~(l+lzIT) 8

on the circle Iz - iAkl = !S, and so an application of the maximum-modulus principle to the function In IIJI (z)(z -lAn)-ll.

< fl, yields lljl' (IAn> I > c I A" IT,

which is harmonic in the circle

Iz - iAkl

where c is a constant. A consequence of (11') is that 1jJ{z) is of completely regular growth. Hence (nIA,,) -+ (dI21T), and so the series (12) is absolutely convergent. We now show that the representation of the function /(1) as the series (12) yields the value zero in the interval (-dI2, dI2). Integrating the function e-/1jJ{z) around the contour r (Figure 10), we obtain 4>. ,

~ ./(1:,,)

lA-ie

I
=

~I r :~;; ci,

dz.

Sec. I]

FUNCTIONS DEFINED BY VALUES ON AN INTERVAL

401

For t < + d/2. the integral along the circular arc tends to zero with increasing N, and hence

f

I+ico

f (t) =

1

21t1

e- t•

+(z)

(t < ~).

dz

I-joe)

On the other hand, integrating around

f+

e- t•

. 1

0= 21t1

For t

>

r 1 (Figure II) we obtain (z) dz.

(r., -d/2, the integral along the circular arc tends to zero, and we have

fe-f. +

.+joe)

1

0= 21t1

(z) dz.

I-joe)

Thus f(/) = 0 for -d/2

< t < d/2.

The theorem has thus been proved. !J

!I e+L'N

e+iN

o

£-LN

e-UI

Fig. 10

Fig. 11

In the proof of Theorem 1, we discarded the non-negative term R

f '\(1) dt o

from the inequality (8)

J•

fIt(t)

o

t

dt > ~ R+.!. InR+ 11:

II

J• 0

IIdt)

t

dt+ 0 (1).

We now take this term into account. We will consider S,.-almost-periodic fUnctions satisfying the conditionf( -x) == f(x), i.e., functionsf(x) such that co

f(x} __ ~ 4".''l"""

Where the a .. are real.

-co

402

APPLICATION OF UNIQUENESS THEOREMS

[APPENDIX II

We denote. by Au the Fourier exponent off(x) nearest to zero, and enumerate the numbers An in order of increasing value ... < )'-k < ... < A-I < Au < Al < ... < A,t < . . .. We use the notation no(t) for the number of changes of sign in the sequence of coefficients ... , a_,t, a-k+l' ... , a_I' ao. ai' ... , a,t, ... corresponding to the numbers An of the interval (-I, t), and write R

No(R)=

f

no/t) dt.

4. If an Sp-almost-periodic function f(x) is zero in the interval ( -d/2, d/2) but not identically zero, and iff( - x) = fix), then the density of the changes of sign No(R) in Ihe Fourier series of this function satisfies the conditioll THEOREM

11m [No(R) -

R-+<»

PROOF.

Let a"a n +1

!!.. R +.!.. In R] > P

'It

> O.

00.

(13)

Then the Laplace transform co

~

a"

cp (z) = ~ z- fA" -00

of f(t) is purely imaginary on the Oy axis and changes sign in the interval (iAn> iA nH ), and so it takes the value zero in this interval. Thus the number of zeros n1(t) of q;(z) in the circle Izl ~ t is not less than the number of changes of sign in the sequence of coefficients {a,,} corresponding to IA"I <; I. Thus

na (t) -

". (t) ~ "0 (t)

+ 1.

If q;(z) ~ 0, there is no loss of generality in assuming that q;(0) :F 0, and from Jensen's formula (7) we obtain 2><

No (R)+ In R >

- 2~

J

In / cp (Re ie ) / dfJ

+ In /cp(O)I·

o

It thus follows from (5) that

No(R) > ~ R - .!..In R+ 0 (1). 'It P This completes the proof of the theorem. Theorem 4 can also be modified to apply to a function of the form co

f(x)

=

f

e itx do (t).

(14)

-co

where 0(1) is a real (unction of bounded variation. An interval on which o(t) is constant will be called an extremal interval if all the values of o(t) in some neighborhood of this interval are either not less than or not greater than the value on the interval.

Sec. I]

403

FUNCTIONS DEFINED BY VALUES ON AN INTERVAL

An extremum point, not in an interval of constancy of O'{I), will be considered as a particular case of an extremal interval. If, at a point of discontinuity, both the left and the right limiting values of O'{t) are extremal, then we will consider that there are two extrema at this point. We denote by no(1} the number of extremal intervals covered by the whole interval ( - t, I), and also use the notation R

No(R) =

f

no/(/) dt.

o THEOREM 5. If a function fix) is given by (14) and is not identically zero, and if it is zero in the interval ( -d/2, d/2), then the density of Ihe sel of extremal intervals of the function O'{t) satisfies the condition'

lim [No(R)-!!... R] >-00.

R -. 00

11:

The Laplace transform off(x) is

PROOF.

()- f

00

'¥

Z -

da (I) It-z'

(15)

-00

It is not difficult to construct a step function 11,(1) such that the extremal intervals of positive length of O'{I) are extremal intervals of 00,(1), and the extremal points of O'{I) coincide with the ends of extremal intervals of a,(t). In addition, this function can be constructed so that

I " (t) -

". (t) I

<e

(-00

< t < 00),

(16)

where E is an arbitrary given positive number. The value of no(l) for 0'(1) is not less than the value of no,.(t) for 00.(1). We now define the function

f

<Xl

(jI.(Z)=

da. (I)

or.>. ,., ak

Jt-z = ~/l.)-z'

-<Xl

-co

where the I~l) are the points of discontinuity of a,(t) and the of 00.(1). We have

J' <Xl

tp

or

(z) _ !Po (z) =

(17)

k

f

at are the jumps

<Xl

d [a (I) - a. (t», It-z

-co

I

a (I) - a. (I) (/I-z)1I

dt

'

-co

(z=x+ly).

(18)

-00

l.A si~ilar, somewhat stronger result was obtained by M. G. KreIn in the theory of COntinuation of Hermite-positive functions.

404

APPLICATION OF UNIQUENESS THEOREMS

Applying Jensen's formula to fP.(Z R

S

[APPENDIX II

+ 6) (6 > 0), we obtain

~

n.1•• (r) ;- nit. (r)

dr

= 2~ Sin 1IP. (Re ce + S) I da -In 1IP. (0)1.

o

(19)

0

where n1,.(r) is the number of zeros and n2,.(r) the number of poles of fP.(z + (j) in the circle Izl .;;; r. All the poles of fP.(z) are on the imaginary axis. and there is a zero between any two poles corresponding to two successive terms in the sequence {aD with the same sign. Thus for t = v'," - (jB, we have ~ •• (r)- nl •• (r)~ -

lno,.(t)

+ I)

and from this result and (19) we obtain 21<

dt

~ In 1IPI (0) 1-- 2~ Sin i!PI (Re"+ 0) 1dO. t'

The inquality

no (t) no. I (t) nO. I (I) ---~---->----;::==-

yt3 +8a

t

thus yields

2..

No(R)

+ In 2~> -

2~ Sin 1IP. (Reie + 0) Ida + In lIP. (0) I.

We now fix a value of 6 so that fP(6) p£ 0, and then, using (18), take the limit for e -+ o. We obtain Is

No (R)

+ In 2aR ~In / IP (0) / - ~ fin / cp (Rete + 0) / dO.

(20)

o

Hence, using (6), which is obviously valid in this case, we arrive at the result

No(R) >!!.. R It

+ 0 (I).

The theorem has thus been proved. The question naturally arises of the possibility of determining the function.f(z) with a given spectrum from its values on an interval (-d/2, d/2), when these values determine the function everywhere. We will limit ourselves to the case when the S.-almost-periodic function I(t),."., ~a~AI:'

is holomorphic in the segment [-d/2, d/2], and we will assume that we know its values in some neighborhood of this segment in the complex plane and that the real sequence {At} is a regular set with density 11 = d/.".. The function CD A) 1 - fA" 8 (A) =

IT ( -CD

Sec. 2]

405

QUASIANAL YTIC CLASSES

is an entire function of exponential type and of completely regular growth, and its indicator diagram is the segment [-d/2, d/2l of the real axis. We use this function to construct a biorthogonal system for the sequence of functions

{e iAtl }. The functions co

f

rp" (z) =

8 (>..) e-'A8

8' (I;",,) (>.. -A,,) d'A

(O;-k=O. 1.2 .... )

o

are holomorphic everywhere outside the segment [-d/2, d/2l, and if C is a contour encircling this segment, then

f ~k(z)eil.p·

2!1

COl

dz = {I 0

for fori

p

= k.

p =1= k.

It immediately follows that a"

J~,,(z)f(z)dz

=

(±k=O. 1.... ).

(01

The contour (C) must be chosen so that it contains the segment [-d/2, dl2l and is contained in that neighborhood of this segment where the function J(t) is given. Thus from the values of fez) in a neighborhood of [-dI2, d12] we have obtained its Fourier series, and we have thus determined the function itself. We note that if the sequence {A k } is such that the function

0(>") 1+1>"1 is in U, then we must obtain the biorthogonal system {CPk(t)} from the formulas (±k =0. 1. 2•... ). -00

If the function L2( -d/2, d/2), then

IS

given on the segment [-d/2, dl2l and belongs to /I

a" = 2~

'2

I.l.m

ff

(t) ,,, (t) dt

(0;- k

= O.

1, 2 .... ).

/I

-'2 2. Quasiaoalytic Classes of Almost-Periodic FUDCtio. In this section, we wiIl also consider Sp-almost-periodic functions having spectra {).n} that belong to some set n without any finite limit points. If we also have lim Ng(R)_O J .... CD R - .

406

APPLICATION OF UNIQUENESS THEOREMS

[APPENDIX II

then we infer from Theorem 1 that if a function of this class is zero on any arbitrarily small interval, then it is zero almost everywhere.6 In other words, if two functions of this class take the same values on any arbitrarily small interval, they are equal almost everywhere, i.e., this class is [I]-quasianalytic. By taking a very "sparse" set n, we can obtain classes of functions Tn in which nonzero functions can not be zero on any interval and also can not tend to zero rapidly when the independent variable approaches any finite value. In [1], Mandelbrojt introduced the special concept of classes of quasianalytic functions for periodic functions. For integrable functions, he introduced the idea of a mean zero of exponential order p, defining this order by

p= lim _-++0

In (-In 6 (a» -Ina

(21)

where t.+_

6 (<<) =

f

I/(t) I dt.

(21 ')

t.

Mandelbrojt's fundamental Theorem C is the following: expressible in the form 00

/(t) --- ~ (an~Cos nit ;=1

•

+bn . sin nit),

has a zero of exponel/tial order p at some pOint, with cont·ergence index a < I, al/d if

If a function f(I), (22)

I

if the numbers n~form a sequence

thenf(t) == 0 almost everywhere. A simple proof of this theorem. based on the application of a uniqueness theorem from the theory of entire functions, was given by the author in collaboration with M. S. Livsic in [1]. This new method introduced the possibility of a significant generalization and strengthening of the theorem. In the article [12] by the author, this strengthening of the theorem concerning quasianalyticity was obtained for the class of almost-periodic functions.' We will describe this result in a somewhat simpler but more complete fashion than in [121. We use the same method here as was used in the investigation of functions defined by their values on an interval. • N. Levinson [I). 'The results of the author and M. S. Li\iic were improved in a paper by I. Hirschman and J. Jenkins [I). A further development of this method was obtained in recent articles by J. P. Kahane [1;2).

Sec. 2]

407

QUASIANALYTIC CLASSES

As before, we will characterize the sequence A the function

= {An} (±n = 0, 1,2, ... ) by

B

N,.(R)=

f -t

"A (t)

(23)

-dl.

o

Let ",(IX) be a continuous, nondecreasing, non-negative function, and let 1/1(0) = O. We will say that f(t) has a right mean zero of order not less than I/I(a) at to if t.+.

f

1/(I)/dt~cl'(II)

(0

< « < «.,>.

(24)

fa

A left zero is defined similarly, and also a two-sided zero. For the function 'f'{Ot) we can take, for example, (I) 'f'{ct) = IlXlft (n = 1,2, ... ), (2) 'f'{Ot) = e-ll/czIP, (3) 'f'{Ot) == 0 for 0 < Ot < Oto. Using the function 'f'{Ot), whose rate of decrease determin(!s the order of the mean zero of the function f(t), we form the monotonically decreasing function IX = 1J(r) (0 < r < (0) defined as the inverse of the r lonotonic function

+

In (II)

(25)

T = - -II - .

A higher multiplicity of the zero to of fit) obviously leads to a more slowly decreasing function 1J(r). THEOREM I'.

Let the Fourier exponents of the almost-periodic function (26)

form the sequence A = {At}. If Ihis function has a one-sided mean zero of order tp( Ot) at some point to and if lim [1tNA (R) - RTJ (R) - In RJ =

then f(l) == O.

B-=+co

-

00,

(27)

Thus if the sequence A is "not too densp.," the functionf(J) can not have a mean zero of high order unless it is identically zero. PROOF OF THE THEOREM. It can be assumed, without loss of generality, that f(l) has a right zero of order not greater than 'f'{Ot) at to = O. We consider the Laplace transform of fit), which is CD

,(z)

f

CD

= .--'1 (I) dt = ~ z ~A,. •

-CD

and estimate the size of the function tp(z) in the right half-plane.

(28)

408

APPLICATION OF UNIQUENESS THEOREMS

[APPENDIX II

In order to do this, we break up the integral in (28) into the sum of two integrals:

• co 'P(z) = I ,-"/(t)dt+ I ,-d/(t)dt.

For Re z

o

> 0, we have

•

•

•

co

II ,-.'/(t)dtl
0

·

M =

sup I/(t) I)

O<'
and so

Setting ex

= 1](X), we have 1jI(rx) = e-a.z =

e-x~(X)

I'P (z) I < (1 + ~) ,-Itt) (:I))

(see (25» and

(.;.;

> 0).

(29)

A direct estimate of the size of the Laplace integral in the left half-plane yields for (30)

x
>

If rp(z) is not identically zero, then rp(z) = zm'1'1(z), where m -1 is an integer and '1'1(Z) is a meromorphic function with '1'1(0) neither zero nor infinite. When we apply Jensen's formula to IP1(Z), we obtain R

NA(R)-

J~?)

2~

dt= ln l'Pl(O)/-

2~

o

J

In/'Pl(Re f8 )/dO,

0

and from (29) and (30) It

NA (R)

> 2~

2"

J

(R cos 111) (R cos 11)

+ In (R cos 1.1») dO

" s

-" 2

+ 2tt1 j'

In I R cos fj 1dO

+ 0 (1).

" Since 1](R cos 0)

> 1](R), we have NA(R)

> ~tt R1)(R)+ In R+ 0(1).

=

0 if condition (27) is to be satisfied. The inverse Laplace transformation yieldsflt) = 0, and the theorem is proved.

It thus follows that tp{z)

Sec. 2]

409

QUASIANALYTIC CLASSES

This theorem can be considered as a theorem on quasianalytic classes, assuming that a class of functions is defined by the sequence A in which the Fourier exponents of the functions lie. The proximity of two functions at a point to is given by the mean zero of the difference of the functions at the point. The theorem means that if two functions of a class are too close together at any point, then these functions are identical. If, in an e-half-neighborhood of to, the functionf(t) is differentiable and (to~t~tO+8,

I/(n)(t>l <mn

and if pnl(to

n= 1, 2, ... )

+ 0) = 0 for n = 0, 1, ... , then Taylor's theorem yields I/(t) 1

< ~i (t-to)"

(to~t ~ to+8)

and we have (0

If P( )

«

< ex < 8).

= ,!!10. (II mn + 1)\ «,,+1 ,

then we can use P(:x) for the function V'
< x< 00)

(31)

for any k > 0, will form the class of quasianalytic functions (11). We also note that if a zero of a function is of exponential order p (see (21» and if the convergence exponent of the sequence A is a, then we have N II.(R) < 1Jg>+< (e > 0) asymptotically, and on some sequence of points r n - CXl we will have TJ(rn) > r;l/(p+l)- •• Thus for a < p/(p + 1) condition (27) will be satisfied, and we obtain the following corollary. COROLLARY. If the exponential order p of a zero of the almost-periodic function (26) satisfies the condition .

a

p> 1- at where a is the convergence exponent of the sequence At then fit)

== O.

This corollary provides the direct transfer of S. Mandelbrojt's theorem to the case of almost-periodic· functions.

410

APPLICATION OF UNIQUENESS THEOREMS

[APPENDIX II

REMARK I. By a slight variation of the proof, we can extend this result to S,,-almost-periodic functions. The inequalities (29) and (30) must be altered somewhat, and the condition (27) must be replaced by

I~~ [TCNA (R) -

RTj (R) - .!.In R] = q

R .... oo REMARK

2.

If the point

Xu

is a two-sided zero of order O(IX), i.e.,

z.,+"

f

~.

If(x)ldx-< /J(cz),

X ~

f

If(x) Idx

-<

(j (<2),

So-a

tIIo

then for

00.

0 we have

and the condition (27) is replaced by the condition

I~~ rN.~ (R) -

R .... oo-

! RTj (R)] = -

00.

We will show that the result I;iven in the corollary is exact in the following sense. Let the sequence A have a density satisfying the condition 0 < 0 < I, let infO'Hl - At) = d > 0, and let -),10: belong to the set A when At is in this set. Then there is an almost-periodic function f(l) with spectrum in A which has a zero of order p = 0/(1 - 0) at the origin. To find this function, we set

!! (I + ~). 00

u (z) =

z

From Theorem (25) in Chapter I, we have the asymptotic relation In 1.1(reit)/ ~

Tt~a

coso (I /J 1- -;-).

(32)

510'2

valid in the whole plane with the possible exception of some £CI-set of values of r. We consider, for I > 0, the inverse Laplace transform

Sec. 2]

411

QUASIANALYTIC CLASSES

and use Cauchy's theorem to obtain I(t)

= ~,

-lr+.

f,., 4~:) + ! f

C (•• r)

lc:a+1

f ,., A~:)

+

-lc:a+1

where C(e, r) is the circular arc Izl = r, Re z Using (32) in (33), we obtain

(r:=r'--'). (33)

V+, ~ E

> O.

1/(t) 1< Me"'-cuP (a = cot (a/2) and M is a constant). Setting r = (I/aa)l/(l-tl), we find that

1/(t) 1< M for some

Cd

> 0,

•

exp

(-call-e).

and thus f(1) has a right zero of order not less than p =

a/(l - a).

We now show thatf(l) is an almost-periodic function. For I> 0, the functionf(t) is the limit for r - 00 of the function I+V

Ir (t) =

.}.,

f ,t. ~) 4

•

I-V

An application of the theory of residues transforms this integral into

~ Ir (t) = ~

p.,. I < ,

1 4' (IAn) ,

i>.,,' +

f

dz ,., 4 (z) ,

0' (" r)

where C'(e, r) is the arc of the circle Izl = r with Re z <: E. The last integral, as can easily be seen from (32), tends uniformly to zero on the ray I ;> 0 for r - 00 and r E £0. Thus f(t) is the uniform limit of the almost-periodic polynomials

A similar method can be used for I < O. Thus f(1) is an almost-periodic function, whose spectrum is in n and which has a two-sided zero of order p = a/(l - a) at the origin.

ApPENDIX

III

COMPLETENESS AND LINEAR INDEPENDENCE ON A FINITE INTERVAL OF SYSTEMS OF FUNCTIONS {e iAtz } 1. Completeness Theorems In Chapter IV, we studied the completeness in the complex plane of systems of functions (ei).tz} with {A k } a sequence of complex numbers. Here we consider the completeness of {e iAtz } on some interval of the real axis. The most detailed results in this direction were obtained by Paley and Wiener in [1] and by N. Levinson [1]. In what follows we will assume, with no loss uf generality, that the system {e iA • X } is considered on the interval (-7T, 7T). THEOREM I (N. LEVINSON). For the system {e iAtz }, in which the Ai; (k = I, 2, ... ) are complex, to be incomplete in [1'( -7T, r.) (p ~ I), it is necessary and suffiCient that there exist an entire function f(A), zero for all Ak (k = I, 2, ... ),

that can be expressed ill the form

f(A)

=

f

eil.zljl(x)dx,

(I)

-It

whereVJ(x) E U(-7T, 1T)(I/p function}.

+ I/q =

I)(forp

=

I, IjJ{x)isabounded,measurable

PROOF. If the system {e iAtz } is not complete in U( -1T, 1T}, then it follows from Riesz's theorem that there is a linear functional operating in the L'P( -1T, 7T) space, not identically zero, that is zero for all the functions of the system {e iAtz }. It follows from the general form of a functional in L'P( -1T, 1T) that there is a function IjJ{x) of U( -7T, 1T), not equivalent to zero, that is orthogonal to all functions of the system {e IAtz }, i.e., such that

f"

eil.k'I'ljI (x) dx =

0

(k

=

1. 2, 3, ... ).

. 1t

Thus f(A) is zero at all the points Ai; (k

=

I, 2, ... ).

REMARK I. From the Paley-Wiener theorem (Appendix I, § 3) for p = 2, the fact thatf().) can be expressed in the form (I) is equivalent to the fact that f().) is an entire function of exponential type a " 1T and belongs to L'P( - 00, (0) on the real axis. 412

COMPLETENESS THEOREMS

Sec. 1)

413

REMARK 2. If we assume that the system {e iAtz } is complete in C( -7T, 7T), i.e., that any function continuous on [-7T, 7T] can be uniformly approximated by a linear combination of functions of this system, then the incompleteness of the system is equivalent to the existence of an entire function fe).), equal to zero for all Ak (k = I, 2, ... ), and expressible in the form

(2) in which a(x) is a function of bounded variation on the interval (-7T, 7T). This result follows directly from the general form of the linear functional in C( -7T, 7T). REMARK 3. If fl is a zero of multiplicity m > I of a function f(A) of the form (I) or (2), then differentiating (I) (or (2» with respect to A we see that tp(x) (the differential da(x» is orthogonal to all the functions

Conversely, the orthogonality of the function tp(x) (the differential da(x» to all functions of this form implies that fl is a zero of multiplicity m of f().). In the sequel theorems will be given concerning the completeness of systems of the form {e iAtz }, but the proofs can be immediately generalized to apply to wider classes of functions consisting of groups of the form

Here each number Ak is to be counted mk times. THEOREM 2 (N. LEVINSON [1]). If the entire function f().) can be expressed in the form (I) with tp(A) E U( -7T, 7T) (or i,.. the form (2», if Ao is a zero off().), and if fl is any complex number, then 11 (A) =

~ ~/(A)

can also be expressed in the form (I) (in the form (2». PROOF. We set

We prove the theorem for functions expressible in the form (2).

d't (x) = do (x)

+ I (iJo -

III

).o>,-GeIII

f

,il.JI do (y) dx.

-It

Multiplying both sides here by eW: and integrating from

f ,~d't (x) = "

-~

I(A)+ I (~- AO>

f

"III

,il).-l,,)111

-7T

to

7T,

we obtain

(f ,Uvw do(y) ) dx.

-.-s

414

COMPLETENESS AND LINEAR INDEPENDENCE

[APPENDIX III

Changing the order of integration and using the fact that

f"

eDoD do (x) =

0,

-"

we find that

"

Sei).md't(X)=~ ~f(A), -"

i.e., that

"

The theorem has thus been proved. This theorem and Remark 3 lead to the following obvious corollary. COROLLARY. If the system {eiA.tZ} is complete in L"( -7T, 7T) or in C( -7T, 7T), and if an arbitrary number n offunctions are removedfrom this system and replaced by n other functions e;I';X (j = 1.2 •... ,n) where 1-'1.1-'2' ••• • 1-' .. are arbitrary complex numbersl not equal to any Ak , then the new system will be complete in the same sense as the original system.

This corollary enables us to assume. without loss of generality. that the set At does not contain the number zero, i.e., that the system {eiA.tZ} does not contain the function identically equal to unity. As usual, nit) denotes the number of points At in the circle Izl ~ t and we write

f R

M.

(R) =

n),

;t) dt.

G

The function NJ.
3 (N.

LEVINSON

such that

lim B-+oo

[1]).

If the

sequence {At} of complex numbers is

IN).(R)- 2R+!P InR} >

(3)

-00,

then the system {eiA.tZ} is complete in L"( -7T, 7T) (p > 1). If(3) is satisfied for p = 00, then {eilts} is complete in C( -7T, 'IT).

If the system is not complete in LP( -7T, 7Tl. then there is an entire functionf(A), equal to zero at all the points At, that can be expressed in the form PROOF.

.

f

I(A)~ t~(x)dx -"

(HX)EL

Q

(-1t, 1t),

~

+;

=

1).

1 We could choose 1'; = A. (j = 1,2, ... ,jk), but then we would have to consider the system consisting of the system {eipp} and the functions eipp, X-eipjs, ... , xJ-eipp.

Sec. \]

415

COMPLETENESS THEOREMS

From this representation of the function, we obtain the inequality «-I

I/(A)I~

f

-C+I"

f + f)eINlII+(x)ldX

e,NlII'HX)ldX+(

-("-I)

(A

"-I

-"

= u+lfl)

and, using the Holdet" inequality, we have

1<-1

t

f

1/(1.)1 ~ e(1<-llItlIIJ'fII-ii(

.!.

1.t(X)llldX)1l

-"+1

!.

-"+1"

1

+ecl·qJ'flI-'[( f

+

-'C

f)II\I(X)r ]ll, dx

a-I

and finally 1

I/(A) 1< Ae"I.'

I fI j-' (,-11 til

+ 0),

(4)

where A is a constant and -"+1"

t

f + f )11\1 (x) III dx ], •

a= [(

'II-a

_K

It is obvious that t5 - 0 when E - O. (This is also true when p == 1.) When we apply Jensen's theorem to I(A) and use (4), we find that

N,(R>.:(,. 2R-

!

...

In R+

~f In I ,-1 1"1+ 01 dfJ+O(I).

(5)

o

Since all the points At are zeros off(A), we have N ,(R) ;> NA(R). In addition, if E > 0 is small enough and R > 0 large enough, the third term on the right in (5) can be made less than any arbitrarily large negative number, and so lim B-+oo

{N).(R)- 2R+!.lnR} =-00. P

Hence the theorem on closure in LP( -'fr, 'fr) follows. The theorem for C( -'TT, 'TT) can be obtained somewhat more simply. COROLLARY.

q 1

P."I~I"I+2p

(:t:: " = 0, I, 2 •... ),

then the system {eil"s} is complete! in LP( -'fr, 'TT). This result is easily obtained from Theorem 3. We omit the proof. Many different criteria for completeness follow from Theorem I. To derive these criteria, we note first ofall thatf(A), expressed in the form (I) or (2), is a I

This theorem was proved.by N. LevinIoa (1).

416

COMPLETENESS AND LINEAR INDEPENDENCE

[APPENDIX III

function of completely regular growth and belongs to the class A (Chapter V, Theorem II). The set of its zeros 1-'1 (j = 1,2, ... ) must therefore satisfy the following conditions: 1. The set {I'/} has an angular density with exponent p = I, and ~(O) is a piecewise-constant function with two different jumps, not exceeding unity, at 0 and1/'. 00

2.

~Ilm ksO

:k 1<00.

3. The limit exists. We call such a set "smooth." For example, the set of even real numbers, having a density, is smooth. A set {A.t} will be called N-smooth if the magnitude of the jumps of ~(O) is equal to N I. The following theorem is obvious.

<

THEOREM 4. If the set of points {At} can not be included in any N-smooth set, then the system {eiJ.,z} is complete in C( - N1/', N1/').

Various sufficient conditions for completeness can be obtained from this result. (6)

<

< < 1/',

where n1(r) is the number of the points At inside the sector Izl r, arg z y then the system {eiA,z} is complete in C( -1/',1/'). • In fact, for every smooth sequence {I't}, from condition I we have

lim lim m (er) - m (r) ~ I,

E~l r~oo

(;-I)r

(1)

<

<

where mer) is the number of points I't in the sector Izl r, larg zl y < 1/'. It is evident that condition (7) also holds for any partial sequence taken from £l-'t} and so when ccndition (6) is satisfied the sequence {At} can not be included in a smooth set. (b) Iflarg At - 1/'/21 T < 1/'/2 and

<

(8)

then the system {eiA,z} is complete in C( - L, L) for any L > O. In fact, any subsequence of an N-smooth sequence must satisfy condition (2). But

• The left-band side of this inequality is called by POlya the maximal density.

Sec. I]

417

COMPLETENESS THEOREMS

Consequently the sequence {I'k} is not included in a smooth sequence, and the system {ej),tz} is complete. For purely imaginary At this theorem coincides with the MUntz theorem concerning the completeness of systems of the form x 1• on any segment a x <; b, b > a > O.

<

(c) If p·t} is the set of zeros of an entire func~ion q;(A) of exponential type, the .....idth of whose indicator-diagram in the direction of the Oy axis is not less than 21T(h,,{1T/2) + h.,( -1T/2) ;> 21T), and if on any straight line parallel to the real axis (1m A = a) the inequality

I,(,,+'a) I>

m

(9)

1

(1+1111)' is satisfied for some m

> 0, then the system {eujr%} is complete in £1'( -1T, 7T).

PROOF. If the system of functions {e iAtz } is not complete, then there is an entire functionf{A), equal to zero at the points At (k = 1,2, ... ), that can be expressed in the form (1). It follows from this representation that ftA) is of exponential type not greater than 7T and is of completely regular growth. The quotient

is a function of exponential type, satisfying on the straight line v = a (v = 1m A) the inequality 1

I1.(" +'a) 1<

M (1 +1" I)"i,

(10)

in which M is a constant. From Theorem 11, Chapter V, the function xCA) is of completely regular growth and its indicator diagram is a segment 1% of the imaginary axis. The indicator diagram of ft).) is the sum of the indicator diagrams of the fqnctions q;().) and X{).) (Chapter III, § 3, Theorem 4). Since the width of the indicator diagram of q;().) is not less than 27T and the type of fO.} does not exceed 7T, therefore the length of the segment 1% is zero and X{).) is of exponential type zero. If X().) has the zero II, then from (10) the function

y. (AHA _1')-1 is bounded on the straight line parallel to the real axis and is thus constant. Since X(A)(A. - 11)-1 tends to zero on this line, we have xC).) == 0, and softA) == O. If X(A) has no zeros, then it is constant and I(A) = C!p(A),

where c is a constant. From (9) we obtain

I/(,,+'a) 1>

em 1 (1+1111)"

(-00<"<00),

418

COMPLETENESS AND LINEAR INDEPENDENCE

[APPENDIX III

and so 00

f I/(u+ La) I" du =

00.

-00

On the other hand, for p

> 2 the representation (I) of f(A) yields4

f I/(u + la) I" du -< ( f 00

"

eOft'I

Hv) III dV)-1

< 00.

-"

-00

Thus a functionf(A) with the properties described in Theorem I does not exist, and so the system {e iAtz } is complete in LP( -7r, 7r). A similar result also holds for the completeness of the system {e iAtz } in C( -7r, 7r). We need only replace condition (9) by the condition

I'P(u+la) I>- m > O.

11m

I" I~ 00

I 'P(u +la)1 =

00

(-oo
1(A) = c'P (A). and so the condition lim,ul_"" I(u ± ia)1 = 00 is incompatible with the representation (2) off().). Thus. from Remark 2 following Theorem I, the system {e iAtz } is complete in C( -7r, 7r). If the condition h",( -7r/2) + h,,(7r/2) ~ 27r in (c) is replaced by a somewhat stronger condition, then condition (9) becomes superfluous. More precisely: (d) If {At} is the set of zeros of an entire function tp{A) of exponential type and lim I.~oo

I 'P (Iv) Ir " I • I > 0,

then the system {eiAtz } is complete in LP( -7r, 7r) (p

(11)

> I).

If the system {e iA ,,-} is not complete in LP( -7r, 7r), then there is an entire functionf(A), zero at all the points At (k = 1,2, ... ) and expressible in the form (I), which consequently satisfies the inequality (4). From this inequality and condition (II), it follows that the entire function PROOF.

1.. (A)

=,I ().) ("A.)

of exponential type is bounded on the imaginary axis, and from Corollary 2 of Theorem 6 (§ 4, Chapter V) it is of completely regular growth and its indicator diagram is a segment 1% of the real axis. The indicator diagram of f(A) is the sum of the indicator diagrams of tp{A) and x().). Therefore the length of 1% is not greater than the width of the indicator diagram of f().) in the direction of • See Zygmund (I), p. 311.

Sec. 2]

419

MINIMALITY

the real axis and is thus zero. Thus X(A) is of zero type, and since it is bounded on the imaginary axis it must be constant. But thenj(A) = CO, ItI~oo

which contradicts (4). Hence the completeness of the system {eiAa2'} in U'( -7T, 7T) follows. 2. Minimality (Strong Linear Independence) of the System {e iAa2'} Of particular interest are complete systems that do not contain superfluous functions, i.e., systems such that one member can not be approximated by linear combinations of the others. In other words, it is a question of systems {iAt%} for which it follows from the inequality

" a~")e4).k'I'~ = 0 lim II ~ n .... oo k&l

that a~n)=o.

lim " .... 00

Here the U( -7T, 7T) norm will be used. A system offunctions with this property (not necessarily complete) will be called minimal. THEOREM 5 (PALEY AND WIENER [I)). For a system offunctions {eiAt%} to be minimal, it is necessary and sufficient that there be an entire function
I'(~)I

EL2(_'It 'It).4a

'

l' 1 +~I

(12)

PROOF. If such a function
hl (x)

f all (l.) e-D.:D dl.,

= 2~

-00

we obtain a system {hk(x)} of functions of L"( -p, p) which are zero outside the interval (-p,p). These functions form a biorthogonal system, i.e.,

J,

{0

hl (x).4).i" dx =

1

-Jf

for k+}, f9r k =}.

Now let the sequence of sums It

~

a}JI).1l1'

11-1

... Similar conditioDa c:an be given for systems {elllo".xe lllo" ••••• x ... -1e lllo"}.

420

COMPLETENESS AND LINEAR INDEPENDENCE

[APPENDIX III

tend to zero in L"{ -1T, 1T). If we multiply these sums by ht{x) and integrate from -1T to 1T, we find that lim,,_ao aj") = O. Conversely, if the system {e iAtz } is minimal, then the function e,Alz is at a positive distance from the linear span of the remaining functions, and thus there is a functional, equal to zero for all these remaining functions, and equal to one for eiAI 2'. By the general Riesz theorem this functional is specified by some function hex) E L"{ -1T, 1T). We have

j ,fA.II"h (x) dx = {O1

-.. The entire function

k =f= 1. k= 1.

..

tp (A) = (A -

AI)

JeD.:J:h (x) dx

is zero for all the At (k = 1,2, ... ) and tp{).)U. - ).1)-1 E L"{ - 00, (0). The theorem is proved. THEOREM 6. For a system {eiAk-.} to be complete and minimal in L"{ -1T, 1T), it is necessary and sufficient that the following conditions hold: 1° the limit

n (1 --f-)

tp(A)= 11m

R ~ 00 I).. I < B

"

exists; 2° tp{).) is an entire function of exponential type with indicator diagram of width 21T;

4°

if x().) is any elllire function

of exponential type zero, then

tp(A)x{A)EL9(-00. 00). PROOF. We first prove the necessity of the conditions 1~4°. From Theorem S, it follows from the minimality of the system {e'Atz} that there is an entire 1T, equal to zero at all the points {At} function tp{A) of exponential type p and satisfying condition (12). Without loss of generality, we may assume that h.,{1T12) = h.,( -1T12). If the degree p of fP{).) is less than 1T, then the function

<

tp (A) ala

(K;- p»). E L' (- 00. 00)

and has degree 1T. Hence, from Remark 1 to Theorem 1, if we adjoin to the system {eiAas} all the functions of the form eip with II any nonzero zero of

Sec. 2]

421

MINIMALITY

sin (7T - p)).., the resulting system is complete. Since the system {e iAtz } is complete, this is impossible, and thus p = 7T, i.e., the width of the indicator diagram of tp()..) is 27T. Condition 3° follows from Theorem 5 and the minimality of the system

{e iAkX } • The function '1'()..) has no zeros different from )..t. If this were not true, further exponential functions could be adjoined to the system {e iAtz } without destroying the minimality of the system. But this contradicts the completeness of the system {e iAtz }. From Theorem II of Chapter V, it follows that when condition 3° is satisfiec! for a function !p()..) of exponential type, this function is of completely regular growth, belongs to the class A, has an indicator diagram that is a segment of the imaginary axis, and can be expressed in the form fP ().) =

ceil..

lim

II (1 -

B-+oo I .." I
;,.A. ).

II:

With no loss of generality, it can be assumed that 1=0, since if !p{)..) satisfies conditions 1~4°, then e-iLAtp()") satisfies the same conditions. It only remains to deal with condition 4°. If for any function x()..) of genus zero we have fP(k)xO.)EL:l (-00, 00). then we obtain a system that is not complete in L2( -7T, 7T) when we adjoin the exponentials {eil'tz} to the system {e iAtz } which is originally complete in L2( -7T, 7T). Here the I't are the zeros of x()..). But this is not possible. Such a function x()..) therefore does not exist. The necessity of conditions 1~4° has thus been proved. For the proof of sufficiency, we note that by Theorem 5 the minimality follows from conditions 1~3°. Moreover, it follows from these conditions that r(i.) is of completely regular growth and that its indicator diagram is a segment of the imaginary axis. If the system {e iAtz } is not complete in L2( - 00, (0), then, from Remark 1 following Theorem I. it follows that there is an entire function I()..) of exponential type p not greater than 7T which belongs to the class L2( - 00, (0) and which is zero at all the points {)..t}. With no loss of generality it may be assumed that its indicator diagram is symmetric about the real axis. The quotient (13)

is an entire function of genus zero. and q()..)X()..) E L2( - 00, (0). But this condition 4°. The system {e iAtz } is therefore complete and minimal In L2( - 00, (0). The theorem has thus been proved. ~ntradicts

422

COMPLETENESS AND LINEAR INDEPENDENCE

[APPENDIX III

From Theorem 6, which is a very general theorem, various conditions of a more special type can be obtained for the completeness and minimality of a system {eiAtZ }. It will be shown in Appendix VI (Theorem 7) that if A,,=k+~(k)

(:±:k=O. 1.2•... ).

(14)

<

where tp(z)e- i '" is an entire function of finite degree (1 'IT and has a bounded absolute value on the real axis, then the entire function of exponential type cp(A)= lim

II (I - i.i.)

B -+ DD I )." I sup ltp(k)1 and M" and m" are positive numbers. It easily follows from this theorem that in this case the system {eiAt:l} is complete and minimal in LI( -77,77). Conditions 1°, 2°, and 3° are obviously satisfied, and we need only verify condition 4°. Let X(A) be an entire function of zero type, and let cP (Il) X(u) EL'l (- 00, 00).

It follows from the Paley-Wiener representation of an entire function of exponential type belonging to LI( - 00, 00) that

,(u-+-Ih)x(u-+-Ih)EL'(-oo. 00).

If the left inequality in (15) is used, then it follows that X(u and so x(u) E LI( - 00, 00). The function 9(u)

± ih) E LI( -

00, 00),

"

= f X'J(t)dt o

is of exponential type zero and is bounded on the real axis. Thus, from the inequality of S. N. Bernstein, it follows that 1p'(u) == 0, i.e., X(A) == O. Condition 4° is thus satisfied and the system {e iAtZ } is complete and minimal. It can also be shown that {ei1tZ } is complete and minimal in LI( -'IT, 'IT) if 1p(z) satisfies the same conditions as in (14) and IAt - k - 1p{k)1 < 1. We do not prove this result here. It is interesting to compare some of the results obtained in this appendix with similar results from Appendix II. From Theorem 5, Jensen's formula, and the inequality (4), it is easy to obtain the following result: If the system {eiAtZ} is minimal in the interval (-'IT, 'IT), then

lim [NA (R)-2R- .!..lnR] =-00.

B-+DD

f/

(16)

Sec. 2)

423

MINIMAlITY

In Theorem I of Appendix II it was established that if

lim [NA(R)-2R-.!lnR]=-00 q

(17)

B-+oo

and {f the sequence offunctions Pn(X)

=

Pn ~ ar:)e,l,,:IJ k= 1

converges in S" to a function equal to zero 011 the segment (-7T. 7T). then lim,,_oo a1") = O. The condition (16) is necessary if the L" convergence of P,,(x) to zero in the interval (-7T. 7T) is to imply limn_co a1") = O. The condition (17) is sufficient that the S" convergence of the sequence Pn(x) to a function equal to zero in (-7T. 7T) should imply lim .. _ oo a1") = O. We note further that from Theorem 3 the condition lim [NA(R)-2R-!lnR+lnR]

B-+oo

q

>-00

is a sufficient condition for the completeness of the system {eiA~} in L"( -7T, 7T).

ApPENDIX

IV

APPLICATION OF UNIQUENESS THEOREMS TO SOME PROBLEMS IN THE THEORY OF DIFFERENTIAL EQUATIONS 1. Completeness of a System of Solutions of a Differential Equation of Second Order in Regions in the Complex Plane In the investigation of the boundary-value problem y"-q(X)y+).9y =O, y' (O)-hy(O)

= 0,

y' (1t)+H"(1t) = 0,

}

(1)

in the theory of differential equations, where q(x) is a real function integrable in (0,11), an essential role is played by the completeness of the system of solutions q{x. Ak ) of this problem in the interval (0, 11). We will assume that in the interval (0, 11) the function q(x) takes values that are the same as those of some entire function q(z) in the same interval. Then the general solution of equation (I) is an entire function of z, and so the solutions cx(z, At) of the boundary-value problem are also entire functions of z. We propose to investigate the completeness of the system of solutions cx(z, At) of this boundary-value problem in a region of the complex plane (in the simplest case, wheny(O) = Y(11-} = 0). We will also investigate the completeness of the system of solutions e(z, At) of equation (I) forA = At with the initial conditions e(O. Ak) = I. e'(O, At) = iAt • where Ak is a sequence of complex numbers. For q(x) = O. these conditions determine solutions of the form e iAtz • The results given here are closely related to the theorems concerning completeness given in § 7 of Chapter IV. For this investigation. we will use "transformation operators" which convert the function eiAz into a solution q{x, A) of the differential equation (I). These operators were introduced by Delsarte in [l) and were subsequently used in several applications in the work of B. M. Levitan [3) and A. Ja. Povzner [I). The latter gave a construction of these operators using Riemann functions. A detailed study of these operators and their systematic application in the theory of differential equations was carried out by V. A. Marcenko {I). To obtain the transformation operators, we follow A. Ja. Povzner and consider the partial differential equation

" - q (y) u = u:=. " un 424

(2)

Sec. I]

425

SOLUTIONS OF A DIFFERENTIAL EQUATION

Th.: solution of this equation for the initial conditions

" Ir - o = ~ (x), and y

> 0 is given byl "(x, y) =

,,(x

+ y)

l'

(x

"~ Ir - o = '-l' (x)

(3)

-=-B

W

~r

J

('Hnv(E, 0, x, y)-,?(~)v~(e, o. x, y)]dE, III-r where l'(~, TJ, x, y) is the Riemann function of equation (2), i.e., the function that is the solution of the equation

+-}

" ""'1'1-"tE

q(1j) " =

0

inside the triangle '1 > 0, ~ + t7 < x + y and ~ - 1J > X - Y for any fixed x, y (y > 0), and equal to unity for ~ + TJ = x + y and for ~ - 1J = X - y. If q(x) is integrable on (0, 1T), then the Riemann function I'(~. '}. x, y) and its partial derivatives are continuous. Starting from (4)', we construct the operator that transforms a solution of the equation y' + ),2y = 0 into a solution of equation (I). Let :Xh(x. ),) be the solution of the equation

«" - q(x)«

+ Aiel =

(5)

0

satisfying the initial conditions «(0, A) = I,

«' (0, A) = h,

(6)

where h is an arbitrary complex number. The function u(x,y, ),) = :lA(y')') cos Ax is the solution of (2) for the initial conditions ~(x)=coIiAx,

'-l'(x) = h cos Ax.

Substituting these results in (4) and setting x = 0, we obtain for the fUlll,tion (lA(Y, ).) the representation

a" (y,

A) =

+~

COli

AY

J

(v (e, 0, 0, y) h --

v~ (e, Q, 0, y)]

COli

I.e dE.

-r

This equation can obviously be transformed into III

«,,(x, A)=cosAx+

f K,,(x, t)cosAtdt.

(7)

o 1 See, for example, A. N. Tihonov and A. A. Samanlul, Etpitlliolu of """Iw_I;(;111 pII,,;('I. Gostehizdat. 1953. pp. 127-130.

426

[APP~.NDIX IV

APPliCATION OF UNIQUENESS THEOREMS

The Volterra operator correspondinfi to

J

KIf] =/(x)+

K,,(x, t)/(t)dt

o

will he called the transformation operator. This operator transforms the(unction cos i.x into a solution ol equation (5) for the initial conditions (6). Denoting the solution of equation (5) for h = by dx,}.). from (7) we obtain

°

x

JV:I(~'

(x. A)=COSAX-+

(8)

0,0, x)cosA;d;.

-x

If s(x,}.) is the solution of (5) for the initial conditions 5(0, ;.) = i., we obtain for s(y. i.) sin AX the expression

=

°

and

5'(0, i.)

x+y

S

+~ f

(y, A) sin)s =

v (E, 0, x. y) sin I.E dE.

x-y

Differentiating both sides of this relation with respect to we obtain S

(y. ).) = sin I.y

+

+Iv: (:,

X

and setting x

0, 0, y) sin I.S dE.

=

0,

(9)

-y

We note that the solution e(x, i.) of (5) that satisfies the conditions e(O, I,) = I, e'(O, A) = iA can be written e(x, i.) = c(x, A) + is(x, }.), and then deduce from (8) and (9) that e(x. ).)=e ih

x

+

;)ei)~+K2(x, Oe-ii~ldE.

J[Kl(X, ·-x

where K1(x, .;) = 1'~(';, 0, 0, x) - v;'(';, 0, 0, x) and K 2(x, .;) = -v~(';, 0, 0, x) x). Finally, writing K(x. = K1(x, .;) Kk(, - .;), we obtain

n

r:,(.;, 0. 0,

e (x, I.) = eiAx

+

x

J

+

K (x, €)

ei)'~ dE.

-x

It is obvious from (10) that x

(X,

i.) = cos i.x

+

fK

(x, ;) cos

i.; dS

-x

and x

S

(x, I.)

=

sin I.x

+

fK -x

(x,

~) sin i·E de.

(10)

II

Sec

SOUl liONS 01, A DIHIIH NIIAI

427

IC}IIA liON

We huvl' thus arrived at TIlHII(I:M I.~ If Ihe fUI/dial/ q(x) iI/ (I) i.I' illlegrah/t' 01/ Ihe il/ten'al (0, rr), then there ex iSIS a kernel K(x, ~), continuous for 0 < x, ¢ < rr, for which the Volterra operator

+f

AI
,m/Illial/ v( \")

(11)

K(x, E)
-ill

1'./

+

y" A9y=0 in/O a solution of (5) sati~fying the same illitial cOntlitiolls as ,vex) for x = O. The Volterra Ilperator (II) will be called a trallsformatiol/ operator (the same as the operator (7»). For our purposes, the following fad is fundamental. If the values of q(x) in (5) are the values of an entire function q(:) in the interval (0, rr), then the kernel K(x, ~) of ,Ire operato,. A[./,(x)] i.l' al/ elllirt' fUIIClioll of both ,'ariah/es. 2a To prove this, it is sufficient to show that the Ril'mann function I'(~, 1/, x,,v) of equation (2) is an entire funelion of all four variables. The change of variables t = X + y, s = x - y reduces equation (2) to j}9u

_..!..

_

ot as and the Riemann function t = ~ + 1/, S = ~.- 1/, T = X

U=

4 q

(t -2 S)

U,

,'a, "I, x, y) is obtained if we make the substitution + y, a = x -

I-

~

t

y in the solution of the integral equation

JJ 8

q (' 2 S) U(I, s) dt ds •

( 12)

G

~

Solving this Volterra equation by the iterative method, we obtain

11= I +u1+u-d+'" +u,.+ ... ,

where U,.(o,

If

I!q«t -

't, S,

s)/2)1

t) ...

-~

t

f Sq('-;")Un_d

( 13)

8

• •

O, 'to

s. t)dsdt.

< M in the region I' - TI IUII(o, 't,s, t)I<M

It·.q(:) i~ an enlire function, then

<; a, Is - 171 <; h, then I s - a 1"1 t - 't I" nil'

(14)

will obviously be an l'ntire fUnctIOn of all its variables, and thus from (14) we sce that the series (13) conve-rges to an entirc function of a, 'T, s, and t. In this case the kernel K(:, :) is therefore an entire function of:: and ~. 11 .. (11, 'T, S, t)

• V. A. Mar~enko [I]. Leont'ev [8] showed that if q(z) is not entire, but only holomorphic in a convex set G, the kernel K(z, n of the transformation operator is holomorphic in the domain z E G, rEG. 2a

428

APPLICATION OF UNIQUENESS THEOREMS

[APPENDIX IV

The resolvent of K(x, ~), which is R(x, -1)

e,

=K(x, e)-K2 (x, e>+ ... +(_1)n+1Kn(x,

e)+ ... ,

also takes values which are the same as those of an entire function R(z, {) in the real domain. Let q;(z) be holomorphic in a region D containing the origin. Then

fez)

=

f•K(z, C)
(15)

o

is also holomorphic in D, and so is

~(z)=f(z)+

f• R(z, c)f(~)dr..

(16)

o For z real, we can integrate along the real axis and obtain y{x) = €p{x). From the equality on the real axis we infer equality in the whole region D. We have thus derived the following theorem. THEOREM

2.

The operators I

+ K and I + R transform every function which

is ho[omorphic in a region D containing the origin into a function ho!omorphic

in the same region, and conversely. We will assume here that q(z) is an even entire function. It follows from (1) and the initial conditions that, for any A, the function c(z, A) is even while s(z, A) is odd. Thus the operator A[q;] transforms an even function into an even function and an odd function into an odd function. THEOREM 3. If q(z) is an entire function and {A,,} is a sequence of complex numbers such that n lim k = I,

n-+oo

n

then every odd function thaI is holomorphic in the simply connected region D, where D colltains the origin and lies in the strip IRe zl < r., can be uniformly approximated by linear combinations of the functions s(z, A,,) in every interior region D' (fY cD). PROOF. We transform fez) with the operator 1+ R. From Theorem 21, Chapter IV, the resulting function can be uniformly approximated by linear combinations of the functions eiAtz in D' so that k ..

IY(z) - ~ ake()."1 < e o

and, since (z) is odd (y{z)

=

l[tp(z) - y{ -z)]), this reduces to kfl

It} (i) -

(zE D')

l ~ ak sin A~ I o

<e

(zE lY).

Sec. I]

SOLUTIONS OF A DIFFERENTIAL EQUATION

429

+ K, we obtain

Applying the inverse operator I

1=..

I/(z)-I~a~(z. 41=)1 o

< 1(1 +PD/)

(zE D,).

where PD , is a constant depending on the region D'. The theorem is proved. LEMMA. Every function /(z), holomorphic in a simply-connected region D not containing any pair of points located symmetrically about the origin, can be uniformly approximated by an odd entire function in every bounded, interior region D' (D' cD). PROOF. The transformation u = Zl yields a one-sheeted transformation of D into some region G not containing the origin, and it transforms D' into G' (e ' c G). The function

F (u) ==

!
holomorphic in G, can be uniformly approximated in G' by a polynomial such that I F{u)-P{u)1 I

<

or, after the transformation to the z plane,

II (z) - zp (zlI) I

< Iz I . I

The proof of the lemma is complete. THEOREM 4.

The set offunctions {s(z, All)}' where Iim}"=I, ft

is complete in 0

-+ Q) " "

< Re z < 11.

PROOF. An arbitrary functionf(z), holomorphic in some region G contained in the strip 0 < Re z < 11, can be approximated with any desired accuracy by an odd entire function X(z), and this latter function can be approximated by a linear combination of the functions s{z, A,.) in a region containing the region G and the origin. The theorem is proved. COROLLARY. If q(z) in (I) is an entire function, then the solutions of the boundary-value problem for y(0) = y(11) = 0 form a complete system in the strip 0< Re z < 11.3 I

It is known that A" - " ..... O. This foDows immediately, if it is recalled that the eigen-

f •

values An are the roots of the equation sin,.A

+

o

X('" 0 sin Ae de

= 0,

and Rouche's

theor.em is used. H the reasoning used here is generalized somewhat, the a>mpleteness of the solutlons can also be established' for more general boundary conditions.

430

AI'I'IICA liON III· IINII}III NISS

'''''.o't! MS

lA1'1'1

NUiX IV

2. 'fht' Intent' Problem for the Slurm-l.Ioutllle .:quatlon We: wnsider Ihe: !lullilion III' Ihe: dilli:rt"nlial e:4ulllion

(0 ./

X ••

I,

t

f Iq

(x) I tlx

<

d

J)

(1)

II

with Ihe: boundury mnJilinn!l y'(O)hy(O)

,0.

(\ ')

Here h lind III Ilre ~omplc:x numbc:rs and q(.\) is a complex-valued funclion that is integrable in (0, I). The sc:l of value:s of). for which Ihilt problem has a nontrivial solution mllke up whal is called ,h" .fpl'(.".,m, ,1',he operator 1.(1') = yW .- q(.\).' for the boundary ~ondilions {h; The: ltpel·trulll of the tlperator (I) 1'''1' the: boundary wn .. ditions {"; h,: will be denllted by .\~(h; hi)' The problem calle:d the invl!rse Sturm-Liouville problem is thaI of fi"d;"K the opera,or L[yl from u knowledge of two of its spel'fru corrl!sponding to different 'boundary ~lIlldiliuns. Tht:' problem of the: unil.Jue determination of an operator from the: spedrll fur different boundary values was first posed in II particular l'use by V. A. AmbafLumian in (I). A detailed study of inverse btlundary rltlhlclll~ Wll .. luallt:' by U. titlr~ in lil. In his wurk on tht" uni4"e determination of lin operah,,, fHlm its boundary-value !lpectra. G. Borg showed thai. if q,(X). (12<.\'), II. h, and h. are reul. then q,(x) = th(x) almosl e:verywhere in 0 " x " I whenevel ,"'vl(h; hi) = S".(II; hi) and SVI(h; II.) = S".
",1.

"I

• M. G. Krein in [6) gave Ilnuther generalization of the uniquenes. theorem that shows that the non-negativ~ function pCx) is uniquely determined by two spectra Sp(h; h.) and Sp(h; h.) of the equation

y" -I- i-.p (.t) y =~ O. We note thai e'luali,>n (I) l',UI be redll~ ..J "I 'hi!> I'llI'm by it ~hallg~ of \'aliables. bUII... re the function pC",) obtained is 1101 only j1O!i.llive but has lin absolutely ~ontinuous d.. rivalive. The Krein result is lherefore stronger Ihlm ,he Borg theorem. • See [II. pp. 385-J\I4. V. A. MarCenko indicates a method "
Sec. 2]

INVERSE PROBLEM FOR THE STURM-LIOUVILLE EQUA nON

431

some simplifications based on the application of elementary uniqueness theorems from the theory of entire functions. We first of aU note that the condition at the origin determines the solution of equation (I) up to a constant factor, and so there can not be two linearly independent eigenfunctions corresponding to a single point of the spectrum. We now define the multiplicity of a point l of the spectrum. We consider the equations

+ AllYl = 0, L (YII] + Ally:) = Y1' L (Yl]

= y" -

(L Iy)

L (y",)

q (x) y).

(2)

+ A2y ", = Y"'-l

We will define the multiplicity of an eigenvalue l to be the largest value n of the number m for which the system (2) has a solution, all of whose components 11,12, ... ,1... satisfy the boundary conditions (1').- A point l of a spectrum with n = I'is called a simple point. In the sequel, we will use the transformation operators already discussed in the preceding section. The main theorem concerning these operators is that, to every differential equation of the form (I) with the boundary condition leO) - hyCO) = 0, there corresponds an operator III

VII) =/(x)+

f K (x, t)/(t)dt 1I

(3)

such that the solution y(x, l) of equation (I), for any value of l and for the boundary condition leO) - hyCO) = 0, can be expressed in the form III

y(x, A)=COSAX+

f K (x, t)cosAtdt. 1I

(4)

o The kernel K,,(x, t) is continuous in the sqllare 0 <; x <; 1,0 <; t <; 1. Substituting this expression for y(x, l) i'l the second boundary condition ,'(1) + htYCI) = 0, we obtain the characteri~;tic equation 1

+ a cos). + f Q(t) cos At dt = 0,

- Asin A

(5)

o

in which a = K,,(I, I)

+ hi and

Q(t)

= K~(l, t) + h1K,,(I. t).

• It is easily seen that tile multiplicity" of a point or a spectrum coinc:ida with tile dimension

or the corraponding invariant silbIpIIce.

432

APPLICAnON OF UNIQUENESS THEOREMS

LEMMA. The multiplicity of a point of the spectrum the corresponding root of the characteristic equation (5).

PROOF.

We write

[APPENDIX IV

Ao is the same as that of

1

+0 cos A+ f Q(t) cos Atdt

(5')

'P(A) = - Asin A

o

and assume that

'P (AO) = 'P' ()·o) =

(6)

Let y(x, Ao) be the eigenfunction. Differentiating equation (I) n - I times with respect to AI, we find that the functions

satisfy the equations (2). It is evident that all these tunctions satisfy the boundary condition at zero, and also, from the relation (6), the condition at x = 1. (-I)" The function zn+l(x) = -n- iJ(A~n satisfies the equation

l-1.

a"

+

(7) L [zn+ll AiZn+l = z" and the boundary condition at zero. In view of (6), it does not satisfy the boundary condition at the point I. It is easy to see that any solution (YI' Y2, ... ,y", Yn+l) of the system (2) for m = n + I, all the components of which satisfy the boundary condition at zero, has the form

Yl = CIZI,

Y2 =

CIZ~

+ C~l'

... , Y"+l

=

c1z,,+l +Z2Z "

+ ... +cn+lzl ,

and so no solution of the system (2) for m = n + I can simultaneously satisfy the conditions at x = 0 and x = I. The system (2) thus has a solution for m = n but has no solution for m = n + I. The lemma is proved. From the expression (5') for 9'
'P(A) =

-A sin A[1 +0 (d-r)]

(8)

outside certain exceptional circles, and so, using Roucbe's theorem, we find that the number of multiple zeros of 9'<).) is finite.? 'Tbe relation (8) also yieldS the simple asymptotic formula A... _ eigenvalues, where f(n) is a complex-valued function tending to zero for n -

+ f(n) 00.

for the

Sec. 2]

INVERSE PROBLEM FOR THE STURM-LIOUVILLE EQUATION

433

We have thus proved that the number of multiple points in the spectrum is finite. and the multiplicity of each of these points ;s also finite. It remains to note that in the self-adjoint case, i.e., for real q(x), hand hl' the spectrum is made up of simple points. In fact, for the second equation in (2) to have a solution satisfying the condition (1'), the right-hand side of this equation must be orthogonal to Yl(X), i.e., the relation

f y~(x)

dx = 0 must hold. But for real

q(x), h and hi' the eigenfunction Yl(X) is real (up to a constant factor), and this equality can not hold. This means that a spectrum with multiple points can not occur. After these general remarks from the theory of differential equations, we now attack the inverse problems. Here two spectra will be said to coincide only if their elements and the multiplicity of each element coincide. THEOREM 1. The function q(x) (0 <; x <; I) and the numbers h, hi and h. are uniquely determined by two spectra of the operator (I) corresponding to two different sets of boundary conditions {h; hl } and {h; hal.

We first prove that the spectrum uniquely determines the characteristic equation. Let the equations

+a

- A sin A

1

1

+ f Q (t) cos At dt == O.

cos A

1

o

(9)

t

+ f Qs(t) cos Atdt == 0

-A sin A+ lltcos A

o

be equivalent. The left-hand sides of these equations are entire functions of exponential type with the same set of zeros, and so they can only differ by an exponential factor. It follows from the asymptotic formula (8) that this factor is unity. and so t

+ f [Q9 (t) -

Q1 (t) I cos At dt == O. o Since the integral in this equation tends to zero if l is real and tends to infinity, we have

(llt - a 1) cos A

t

as == a, and

f [Qs (t) -

Q, (t) I cos At dt == 0

o

and so Q.(t) = Ql(t). We now show that, for given h, two spectra for the same operator (I) determine hi and". uniquely. Actually, the spectra uniquely determine the numbers , Q,(I)==K.(l, 1)+h1K.(l. I). a 1 == K(I. 1) +h" llt == K(l.

l)+hs,

,

Q.(I)==K.(l. 1)+hsK.(l. I).

434

APPLICATION OF UNIQUENESS THEOREMS

[APPENDIX IV

and so also the numbers

h 1

= Qt
and

10_ '"11

= Qt
Now let the spectra {At} and {/At} of the operator

Y' -ql(X)y+).9y = 0

(10)

for the boundary conditions {h; h,} and {h; ha} coincide with the spectrum of (11)

for the boundary conditions {g; gl} and {g; gz}. Then, from what has been proved, gl := hI and gz := ha· Let y(x, A) be the solution of eq uation (10) for the initial conditions y(0) := I, y'(O) = h and let z(x, >.) be the solution of (11) for the initial conditions z(O) = 1, z'(O) = g. We substitute y(x, A) in (10), z(x, A) in (II), multiply the first identity by z(x, A) and the second by y(x, A), subtract the second from the first, and integrate the resulting relation from zero to one. We obtain (y'(x, A)Z(X. A)-y(X. A)Z'(X, A)]/! 1

+ flq,(X)-qg(X»)y(x.

A)Z(X. A)dx=O.

o If A is one of the points of the spectra At or /At, then from the relations hI = gl' ha = g2 the first term in this identity is equal to g - h. Thus the entire function 1

9{).)= g--h

+ f (ql (x)-qg(X) )y(x,

).)z(x. A)dx

(12)

o is zero for all the At and /ATe, while the multiplicity of any zero of tp(l) is not less than the multiplicity of the corresponding point of the spectrum. We will show that the rate of growth of tp(l) is insufficient for it to have the indicated zeros without being identically zero. In fact, the functions y(x, A) and z(x, ,l) can be expressed in the form ~

y(x. A)=cosb+

f K,,(x, t)cosAtdt,

(13)

o ~

z(x, A)=cosb+

f Lf/(x, t)cos)'tdt,

(14)

o and so it easily follows that tp(A) satisfies

I"'().) I < Mel 11.. 11 in the whole complex plane for some constant M.

(15)

Sec. 2]

INVERSE PROBLEM FOR THE STURM-LIOUVILLE EQUATION

435

From previous results, we know that the spectra {Are} and {,Ill,} coincide with the set of zeros of functions IPl(A) and IP2().) of the form (5') satisfying the asymptotic relation (8). Thus tp().) is zero for all the zeros of the function CPI

(A)cp2(A) = 1.. 2 sin 2 A[1 + 0

(m) ].

(16)

where the asymptotic relation (16) holds outside exceptional nonintersecting circles with centers at the zeros of this function. From (15) and (16), we find that the entire function

e (A) =

~

'1 (A.)(A.)CPJ (A.)

satisfies, outside the exceptional circles, the relation 8(1..)= 0

C~llI)

and so the maximum-modulus principle yieldsi' 0()') We thus have the identity

== O.

1

g-h+

f [QI(X)-Q2(X»)Y(x, A)Z(X, A)dx:eO.

(11)

o From this it is not difficult to prove that g = hand ql(X) = q2(X) almost everywhere. From the representations (13) and (14), it follows that y(x, A)Z(X, A)=COs2 ).x ~

+ f K,,(x,

~

t)C0sAtc0SAXdt+

o

f L,(x, t) cos AtcOsAX dt 0

~

+f o

~

f K,,(x, s) L,(x,

t)C0sAscoIAtdtds.

0

Extending the range of the functions K,,(x, s) and L,(x, I) by specifying that they are even, i.e., so that K,,(x, -t) = K,,(x, t) and L,(x, -I) = L,(x, t), we can write 1 y(x, A)Z(X, A) = 2+col2Ax

f

ID

+~

IK.(x, t)+L,(x, t»)COSA(X-t)dt

-~

+! j j -~

K,,(x, s)L.(x. t)coll..(t-s)dtds

(x~O).

-QI

~If it is given that ¥(1) is zero, not for all the zeros of the product 9',(1)9',(1), but for all zc:ros except one, i.e., the spectra do not correspond exactly but are the same except for : : POint, then ¥C1X1 - A.> divides into 9'l(l)lp.(l) and we again obtain ¥(1) • O. This fact ws that some strengthening of the theorem is possible.

436

APPLICATION OF UNIQUENESS THEOREMS

[APPENDIX IV

By changing variables, we obtain

fIII

1 +"2cos2b+ 1 y(x, A) z(x, A) = 2' 2'1

K(x, 't)COS2A'td't,

(18)

-Ill

where 2 [K,,(x, x-2't)+L,(x, x-2't») III

f

+

K(x, 't)=

K" (x, s) L, (x, s -

2't) ds for 't> O.

-Ill+~ 411

f K,.(x, s) L, (x, s-2't)ds

for 't

< O.

-411

After the change of notation ql(X) - q2(X) = rex) [r( -x) = rex)] and the substitution in (17) of the expression in (18) for y(x, A.)z(x, A.). we obtain g-h+

1

f ~ fri

~

r(x)dx

o

1

+

1

r(x)+

-1

fK

(y. x)r (y) dY]COS 2AX dx = O.

III

The last term on the left tends to zero when A. Therefore g-h+

~

00

along the real axis.

1

f

(19)

r(x)dx=O

o

and 1

1 "2r(x)+

S K(y. - x)r(y)dy=O. :1J

By substituting r(1 - x)

= lP(x) and y =

1 - t in this equation, we obtain

III

,(x)+2

f K(I-t,

l-x),(t)dt=O.

o This is a homogeneous Volterra equation, and so Ij1{x) and h = g. The theorem has thus been proved.

== O.

Hence rex)

== 0

ApPENDIX

V

REPRESENTATION OF A POSITIVE ENTIRE FUNCflON OF EXPONENTIAL TYPE AS THE SQUARE OF THE ABSOLUTE VALUE OF AN ENTIRE FUNCflON

The starting point of the results described in this appendix is the following Fejer-Riesz theorem: A non-negative trigonometric polynomial fJ

T(x)=~c~U:.o

(T(x)~O. -00

-fJ

< X < 00)

can be expressed in the form T(x) = I S(x)

r

(-00

< x < 00).

where

The trigonometric polynomial S(x) can be chosen so that all its zeros are in the closed upper half-plane. We note that, in this representation, the trigonometric polynomial S(x) can be replaced by the polynomial e- in"'/2S(x), where the degree of the latter is half the degree of T(x). In [5], M. G. Krein generalized this theorem to bounded entire functions of exponential type, showing that an entire function of exponential type k, bounded and non-negative on the real axis, can be expressed as the square of the absolute value of a function of exponential type kl2 with its zeros in the closed upper half-plane. Krein's proof was based on the approximation of a bounded, entire function of exponential type by Levitan trigonometric polynomials (Appendix I, § 4). N. I. Ahiezer, in [I], further generalized this theorem, giving the following criterion for the expressibility of an entire function of exponential type as the square of the absolute value of another entire function. THEOREM 1. For an entire function f(z) of exponential type to have the representation (1) /(x) = I, (x) r~.

437

438

REPRESENTATION OF A POSITIVE ENTIRE FUNCTION

[APPENDIX V

where !p(z) is all entire function of exponential type kj2 with zeros in one of the half-plalles 1m z ;;;. 0 or 1m z <;; 0, it is necessary and sufficient that fez) belong to the class A and be non-negative on the real axis. PROOF. The necessity is obvious. The fact that f(x) must be non-negative follows directly from (I). Furthermore, the entire function g(z) of exponential type with its zeros in a half-plane must be of class A (Chapter V), and so fez) also belongs to A, since its zeros are those of tp(z) and their conjugates. We now prove the sufficiency of the conditions. A function fez) of exponential type can be expressed in the form co

fez)

=

Z2fJe!aH2bll

(I -

a

~)

e a".

(2)

1

The zeros of fez) on the real axis are of even multiplicity, since f(x) ;> O. We denote the zeros of f(z) in the half-plane 1m z ;> 0 by a~. In this enumeration of the zeros, each zero in the open half-plane 1m z > 0 is repeated a number of times equal to its multiplicity, while each of the zeros on the real axis is repeated a number of times equal to one half its multiplicity. We write (3)

and (4)

It is evident that fez)

~ (z),(z).

=

(5)

We will show that tp{z) is of exponential type. From Lindelors theorem (Chapter I, Theorem 15) and the fact that fez) is of exponential type, we have

~ a~ I<M

Since

(M a constant ).

(6)

10" 1< r

(7)

it follows from (6) that

I

~ Re ~J < Ml

la" I
I~

1~I
a~,

(M1 a constant).

we have

Reali =

iI

~

1~I
Re a~

I< ~

Ml'

APPENDIX V]

REPRESENTATION OF A POSITIVE ENTIRE FUNCTION

439

and using (7) we obtain

I ,~ -i-\<M a"

(Mil a constant).

Il

(8)

I a"l
We also have

n, (r)

1 - . nf (r)

lim - - = - 11m --<00. r 2 r-+oo r

r-+oo

(9)

From (8) and (9) and Lindelof's theorem, it follows that tp(z) is of exponential type. It remains to prove that the exponential type of q7(z) is k/2. The function fez) can be expressed in the form ! (z) = ,!I(z) Z(z) where

z(z) =

~i (1 - a~)( 1-

-l)-

1•

Lemma 5, Chapter V, yields the result that x(z) has completely regular growth in the upper half-plane and has an indicatot equal to zero. From Theorem 5, Chapter III, concerning the addition of indicators for products of holomorphic functions when one of these functions is of completely regular growth, we have h,(6) = 2h. (6) = 2h, (6) (0 < 6 < 1t). Similar reasoning applies in the lower half-plane. Thus hf (6) = 2h, (fJ)

(0

-< fJ < 21t),

and the degree of tp(z) is one half that of fez). The theorem is proved. Another generalization of the Fejer-Riesz theorem can be obtained by considering, instead of finite trigonometric sums, infinite series of the type 00

!(x)=~a"ei>.,.1II

00

(IA"I~6, ~la"l
(10)

1

-00

Every such series obviously represents an entire function of exponential type, the conjugate diagram of which coincides with the smallest segment of the imaginary axis containing all the points iA". It follows from the preceding theorem, of course, that if fix) > 0, then it can be expressed as the square of the absolute value of an entire function tp(z) with degree one half that of fix) and with zeros in one half-plane, but it still remains to prove that 9'<x) is also a function of the form (10). THEOREM 2. Let fix) be of the form (lO) and of exponential type fl., and let f(x) ~ m > O. Then there is a function 9'<x) of the form (lO), with exponential type fl./2 and with zeros in a half-plane, such that l

!(x) = Icp(x) 1'. 1

See B. Ja. Levin [13J.

(11)

440

REPRESENTATION OF A POSITIVE ENTIRE FUNCTION

[APPENDIX V

PROOF. From Theorem I, we know that a function q3(z) of the form (4) is an entire function of exponential type tl/2, satisfying (II), with zeros in one half-plane 1m z > O. It is only necessary to prove that it can be expanded in an absolutely convergent series of the form (10). The basis of the proof is the following theorem proved by Calderon and Pitt.2 If \fI(x) can be expressed by the absolutely convergent series

.} (x)

=

co

~ Cke',A,l",

(12)

-co

in which the At are real, and if the closure of the set of values of tp(x) lies in the region in which the function cI>(z) is holomorphic, then qJ(x) = cI>[qJ(x)] can also be expressed as an absolute~v convergent series of the form (12). In particular, it follows from this theorem that

-co

-00

The function rn

C(z) = 2 ~ bkeiIJ.k

Z

k=O

is holomorphic and bounded in the upper half-plane and I

Re ~ (x) = 2' In I(x). It thus follows from (II) that Re {(x) and In l~x)1 take the same values. We

will show that

~z)

and fi(z) =

et(z)

are essentially the same function. For this we note that ~z) is of exponential type and bounded on the real axis. Therefore <ji(z) is of completely regular growth (Chapter V, Theorem II) and

h.

(6) =

~ I sin 6/.

We know that ~z) has no zeros in the upper half-plane, and so the indicator of the function is zero in the upper half-plane. The function D(z) is bounded in the upper half-plane, and the indicator of the function -

Hz) = D (z) ,-1 (z)e

-i~. 2

is therefore not positive in the upper half-plane. 2

See Gel'fand, Ra'ikov and Silov (1], 129, Corollary 2 to Theorem 2.

APPENDIX V]

REPRESENTATION OF A POSITIVE ENTIRE FUNCTION

441

Moreover, on the real axis we have

I Hx)/ =

l,

and consequently, from the Phragmen-Lindelof theorem (Chapter I, Theorem 22), (1m z:;;:" 0). !Hz)! ~ I The function ~(z) can be continued by symmetry to the lower half-plane, giving an entire function without zeros. But the function -; In ~(z) satisfies all the conditions of Cebotarev's lemma (Chapter Y, Lemma I). Therefore and -

C,.)-i

(~+II).-i~

cp (z) = e 2 Again applying Wiener's theorem, we obtain co

(~!d1r! -CIQ

•

< 00).

We now show that IlCtl <; A/2. It is not difficult to see that the points ilC" are singularities of the function associated with ~z) in the Borel sense. But the conjugate diagram of cp(z) is the segment [-i A/2, i A/2] of the imaginary axis, and so IIC"I <; A/2. The theorem is proved. It should be noted that this theorem can not be applied to products of almost-periodic functions with bounded spectra. For if it is not required that the Fourier series of the entire, almost-periodic function f(x) of exponential type be absolutely convergent, then a positive, almost-periodic function f(x) with bounded spectrum can be constructed for which the corresponding function q;(z) is not almost periodic.3 If it is required that the Fourier series of f(x) be absolutely convergent, then as was shown in Theorem 2 the function f1!{x) is not only almost-periodic but has an absolutely convergent Fourier series. It is interesting to note that there are finite positive sums of the form (12) that can not be expressed as the squares of the absolute values of finite sums. It can be proved, in particular, that for c> 2 and irrational A, the almostperiodic polynomial P (x) = cos x cos AX can not be expressed as a product of almost-periodic polynomials.· A theorem similar to Theorem 2 is obtained when, instead of entire functions of exponential type expanded in trigonometric series. we consider functions expressed as Fourier integrals.

c+

I I

+

An example of such a function is given in Appendix VI. We, of course, exclude the trivial polynomials ell'll.

442

REPRESENTATION OF A POSITIVE ENTIRE FUNCTION THEOREM

3.

Iff(x) is expressible in the form k

k

!(x)=~+

f ei~(A)dA

fl~(A)ldA
(

a constant)

-k

-II

and iff(x) ;;;. q

[APPENDIX V

> 0 for

-

00

< x < 00, then lex) = Iq> (x) I~,

where !p(x) can be expressed in the form II

2

q> (x) =

~1

+ f eiUX (A) dA II

-2

This theorem is proved by repeating, step by step, the proof of Theorem 2, but using the following theorem due to Wiener and Pitt.4a If 00

f ei).aI~(A)dA,

f(x)=~+

-00

and Cl>(z) is holomorphic in a region containing the closure of the set of values of fix) (- 00 < x < 00), then the function [f(x)] can also be expressed in the form 00

-00 00

( f 11\11 (A)I dA <

00,

~1 = ~ N) .

-00

A more general result is obtained if a theorem due to Wiener and Pitt [1] concerning functions expressible as Fourier-Stieltjes integrals is used. We will only give the result without proof. ~ THEOREM

4.

Let II

f (x) =

f

ei),a, da (A),

-Ie

where o(A) is a function of bounded variation in [-k, k J. Let o(A) = a.(,l) + ".(A) + aJA), where a.(A) is the absolutely continuous port of the function o(,l), a,(..1) is its function of Singularities, and aJA) is its jump function . ... See Gel'fand, IWkov and Silov [1], 117, CoroUary 2 to Theorem l. • See B. Ja. Levin [13].

APPENDIX V]

Then

ifI(x) > m > 0 and II

II

f Ido, (A) I< If e

ilal dOtJ (A)

we have where

443

REPRESENTATION OF A POSITIVE ENTIRE FUNCTION

-II

I,

-II

I(x)

= I, (x) ra, II

"2

, (x) =

f

eila d't (A),

"

-2" with T().) of bounded lJariation on [-kI2, kl2l. Moreover, if a,().) = 0, then T,().) = 0, and if a.(l)

= 0, then T.(l) -

O.

ApPENDIX VI

ALMOST-PERIODIC FUNCDONS WITH BOUNDED SPECTRA

In § 2, Chapter VI, we gave some results concerning the distribution of the zeros of almost-periodic functions with bounded spectra. In 1948, M. G. Krein and the author carried out a much more complete investigation of the zeros of these functions and obtained a precise characterization for the set of zeros of functions of the class [~J. These results were published without proofs in [1]. Here for the first time we give these results with pro.)fs, basing our discussion on the contents of § 2, Chapter VI. We will assume (with no loss of generality) that a function of the class [~J is of degree ~ in the upper and lower half-planes.

1. An Expression for the Real Part of a Zero of a Function of the Class [~] It was proved in Chapter VI (Theorem 3) that all the zeros of a function of are in some strip Iyl < h. We denote by n(t), for 1 > 0, the number of zeros of a function f(z) of [~] in the rectangle 0 < Re z <; t, Iyl <; h, while n(/) for t < 0 will be minus the number of zeros in the rectangle t < Re z ..;; 0, [~]

Iyl
1. For functions of [~], we have

n (t) =

:

t

+

III (t),

where £0(/) is abounded fW1ction, almost-periodic in the sense of v. v. Stepanov l (for any p > 0), with the numerical module of this function included in the numerical module off(z). PROOF.

We start from (6.16), § 2, Chapter VI: B

2'1tn(t)

= 2At-'PB(t)+'LB(t)+ Re

-H

where

B

c=Re

i I'

(s)

I(s) d8

-B I

+ rI'(t+,s) I(t+ ds c,

See B. M. Levitan [I].

444

Is)

(I)

Sec. I]

44S

REAL PART OF A ZERO

and CPH(t) and tp_H(t) are almost-periodic functions. If the sequence {h .. } is such that {j(t + h n )} tends uniformly to f(t) on the whole axis, then tp ±H(t + h..) 'P±B(t) (Chapter VI, proof of Lemma 4). From a well-known theorem of Favard,· it thus follows that the modules of the functions'll ±B(t) belong to the module off(t). In § 2, Chapter VI, the boundedness of the function

=:

B

(1)1

f I(t(t++ I'

Re

(t) =

Is) IS) ds

-B

was established. We now show that this function is Sp-almost-periodic with a module belonging to that off(t). Writing M=supl/(x+ly)1 in the strip -

00

< x < 00, Iyl (; H + b, we will have

J

If'(x+iY)I=12~

l(x+c?+C)dCI<

~

(lyl
It I",a

Similarly, for

Iyl (;

H, we have

II' (x +ly+'t)- I' (x+ ly) / (2)

I/(x+ly+'t)-/(x+ly)/.

sup

-00

< III < 00

1111
The intervals «In

-a,

(In

+ a)

(--+- n = 0, 1, ..• ),

where ex.. is the real part of the zero an of fez) and b is positive, will be called "exceptional" intervals. It follows from Lemma 2, Chapter VI, that the length of the intersection of the exceptional intervals with any interval of unit length does not exceed 2(k + 1)b, where k is the number occurring in the statement of the lemma. Let 7' be an arbitrary real number and (x, x + I) an arbitrary unit interval. Let £8..,. be the set of 7' E (x, x + 1) for which either t or t + I belongs to an exceptional interval. Obviously rnesEo, ~<4(k+ l)a. If t E E~.", then from Lemma I, Chapter VI, we have If(t + ;y)1 ;> m > and If(t + T + ;y)1 ;> m > 0, and from (2) we obtain

°

Iy) _I' (1+ Iy) I 1 /'(t+~+ Ilt+~+'Y) l(t+'Y)

<;,!!

sup - 00<1<00

lyl
• See B. M. Levitan [I).

I/(t+'t+'y)-/(f+'y)/,

(3)

446

ALMOST-PERIODIC FUNCTIONS WITH BOUNDED SPECTRA

where M = sup I/(t + iy)1 for - ex> < t < ex> and Iyl " H sup Iw1(t)1 = MI' and using (3) we obtain

[APPENDIX VI

+ 6.

We write

,,+1

DCD I (I +'I:)-CDdl)D: =

sup -00

sup

f

< ,. < 00 "

1CD I (I+'I:)-CD I (I) I' tit

1/(1 +'1:+ ly) - /(1+ ly) 1+ 4 (k+ 1)8M••

-00<1<00

lyl
We thus have proved that with an arbitrary E > 0 is associated a number is an Sp-translation of the function wl(t). Consequently w1(t) is an (S•• E)-almost-periodic function with a module belonging to the module of f(t). It remains to remark that

1/

> 0 such that any 1/-transiation of f(t) 1

8)(1)= 2K [CD.(I)+'a(I)-'_H(t)+cl.

The theorem has thus been proved. The conversion of the formula to one for n(t) yields the theorem. THEOREM 2. Let {ai:}~CD be the sequence of all the zeros ofthefunctionf(z) of the class [d), enumerated in the order of increasing real parts. Then

(4)

where q(x) is a Bohr almost-periodic function such that its smallest numerical module, containing all the products formed by multiplying d/.". with every Fourier exponent of q(x), is contained in the module off(z). PROOF. The function u = n(t) is piecewise constant, nondecreasing, continuous from the right and takes only integral values. Its inverse n_1(u) can be determined, and this inverse is continuous from the right. nondecreasing, and constant between successive integral values of its argument (Figure 12).

u=n(tH)-Ar

u

u=-n(t)

t Fig. 12

Sec. )1

447

REAL PART OF A ZERO

We form the function v'T)

(I)

=

n (I

+ 't) -

b.

-;t 'to

Solving this relation for t, we obtain the inverse function

V<~)1 (u) =

n_ 1 (u

+ ! 't) -'to

We choose or so that it is an e-translation of the SI-almost-continuous function = tl.or/7T is an integer. 3 Then V(T)(t) - n(t} is an integer. On the other hand, Theorem I yields w(t) and such that q a1+1

f

a1+1

IvlT)(t)-n(/}ldt=

x

f

1(I)

(t+-'}--(I) (t) 1 dl

< s.

a1

This integral is equal to the sum of the areas of certain ,ectangles (see Figure 12) the heights of which are integers, and so the bases of these rectangles are not greater than E. We consider t as a function of u, obtaining

I'TJ(~)l(U}'- n_ l (u) 1< s Introducing the function qJl(U) !(jll(U+

= n_l(u} -

(-00

< u < (.JO).

(7T/tl.}U, we have

)I

~ 't)-(jldU =In-l (u+ ~

,) -

't-n_d u )\ <e.

{5}

We now consider the expression for Re ak • From the definition of n(t) we have n(Re ak) = k, or for the inverse function Reak

=

n_ 1 (k)

=

-i k+(jll (k)

In view of (5), to every positive integers v = tl.or/7T such that

I(jll (k + v) -

CPl (k)

E

(-+- k =

0, I, 2, ... ).

there corresponds a relatively dense set of

I< e

(-+- k = 0, I, 2, ... ).

Interpolating with a linear function qJl(k) between successive integral points, we obtain a continuous, almost-periodic function !p{x), and Rea1c= ~ ki-(jl(k}. We further note that for any E > 0 we can find a 15 > 0 such that if or is a b-translation of w(t}, then tl.or/7T is an e-translation of !p{x). Hence the numerical module ofthe function !p{x) multiplied by tl./7T belongs to the numerical module of w(t}. The theorem has thus been proved. • I t is not difficult to show that there is a relatively dense set of such e-translations, and that this set determines the module of the function up to terms that are multiples of ~ (see B. M. Levitan [1D. .

448

ALMOST-PERIODIC FUNCTIONS WITH BOUNDED SPECTRA

Since all the zeros of a function of the class H, it follows from the representation (4) that

COROLLARY.

strip Iyl

OS;;;

ale =

i

[APPENDIX VI

[~]

lie in some

k+ .~(k),

(6)

where ?P(k) is a bounded function. It is interesting to note that there are functions of the class 1m ?P(k) is not an almost-periodic function. The function

[~]

for which

g(z) = sin z sin A(z-l)

with), irrational is an example. In fact, the function 'PI(k) = 1m Ok in this case takes only the two values and 1. For e < 1, every e-translation of lfl(k) must be simply a period of this function. Let the integer T be the period of 'PI(k). Then all the zeros Ok with subscripts k = nT + r (±n = 0, 1,2, ... ) are zeros of only on(" of the functions sin z or sin ).(z - i). Thus the set of points Ok can be divided up into T classes (r = 0, I, . : . , T - I), each of which consists of t.le points of the form m'TT or of points of the form m'TTI). + i (±m = 0, 1,2, ... ). Let Pl be the number of classes of the first type and Pi the number of the second type (PI + P2 = T). Let T" be a number such that nT = neT,,). Then it is obvious that

°

Tn

=

nPITt

+

a 1 (T,,)

=

np9 ~

+

a2 (Tn),

where 0 OS;;; el(T..) < 7f, 0 ~ E2(T..) < 'TTl).. Dividing this equality by nand taking the limit for n -+ 00, we obtain _Pt PI-T'

i.e., ). is rational and the assumption is contradicted. Thus 'Pl(k) can not be almost-periodic. The following theorem is of interest in connection with this example.

If the sequence of zeros of an almost-periodic function of the satisfies the condition

THEOREM 3.

class

[~]

(±k=O, I, 2, ... ),

= (AI'TT)k + ?P(k), where ?p(x) is almost-periodic. PROOF. We denote the rectangle x = ±tJI2, Y =

then ak

±(H + tJ) by I, and

consider the integral

1 f,,(z+ik+'(k») zdz, ,(z+ X-k +.' (k)

'PI (k) = 2n1 •

(1)

11:

)

(1)

Sec. 2]

CHARACTERIZATION OF A SET OF ZEROS

449

in which 'P(k) is the almost-periodic function equal to Re arc - (."./I1)k. It is evident that the rectangle (/) contains one simple zero of the function f(z + (."./I1)k + rp(k», and so 'Pl(k) = i 1m arc. From Lemma I we havel!(z + (."./I1)k + rp(k»1 m > 0 on the contour of integration (/). Moreover, since the numerical module of rp(k) multiplied by 11/.". belongs to the numerical module of f(z), to every number a > 0 we can 0 such that every 'I-translation of f(z) multiplied by associate a number 'I 11/.". will be a a-translation of rp(k). Let E O. We choose a > 0 and 'I > 0 so small that for any 'I-translation 'T = (."./I1)q (q an integer) of the functionf(z) the inequality

>

>

>

I' (z + "+ -i k+ , (k + q») I' (z + i k+ , (k») • < 2(H+2o)1 ) ) Iz+ik+'~ ( ( Iz+"+:k+'~+~ holds. From the integral representation of 'Pl(k) we obtain

1< '"

ICPt (k+q)-CPl (k) Thus 'Pl(k) is an Jlmost-periodic function, and so tp(k) = rp(k) + 'Pl(k) is also almost-periodic. The theorem has been proved. We note that here also we can find an 'I> 0 for any E > 0 such that an arbitrary 'I-translation of f(z) multiplied by 11/.". is an E-translation of 'Pl(k). Hence the module oftp(k) multiplied by 11/.". is contained in the module off(z). 1. A CbaraeterizadoD of the Set of Zeros of FUDCtiOD of the Ous [11]

aD

Almost-Periodic

DEFINITION 1. A number 'T is called an E-translation of the POUrt set {a} if there is a one-to-one correspondence between the points of the set {a + T} and the points of {a} such that the distance between corresponding points is less than E. DEFINITION

to every

E

2. A set {a} of points is called an almost-periodic point set if

> 0 there corresponds a relatively dense set of its e-translations.

LEMMA 1. The set of zeros of an almost-periodic function of the closs [11] is an almost-periodic point set. PROOF. Let the number E (0 < E < I) be given. We choose 6 > 0 small enough so that 2k6 < E, where k is not smaller than the number of zeros in the rectangle t x t + 1, Iyl H. Then the diameter of every region covered by the exceptional circles of radius 6 with centers at the zeros arc of /(z) does not exceed e. From Lemma 1, Chapter VI, there is a number m" > 0 such that

< <

<

I/(z) I outside such regions.

> Ina> 0

(8)

450

ALMOST-PERIODIC FUNCTIONS WITH BOUNDED SPECTRA

[APPENDIX VI

<

Let T be an (m~/2)-translation of f(z) in the strip Iyl H + (). Then from (8) the function f(z - T) has no zeros outside the exceptional regions. Thus from Rouche's theorem the functions f(z) and f(z - T) have the same number of zeros inside each of these regions. Having established a one-to-one correspondence between the set {Ok} of zeros of f(z) and the set {Ok + T} of zeros of f(z - T) inside every such region, we find that T is an e-translation of the set of zeros of f(z). Therefore each (m~/2)-translation of f(z) in the strip Iyl H + () is an e-displacement of the set {Ok}. We note that if OJ; can be expressed in the form

<

~=~+t~

~

where c is a constant and tp(x) is almost-periodic, then the set {Ok} is an almostperiodic point set. The example in § I shows that such a representation does not exist for every almost-periodic point set. LEMMA

2.

Let tp(k) be a bounded function, let the points ak

=ck+t(k)

form an almost-periodic point set,4 and let

ft(l- :J.

f(z)= P.V.

(9)

-00

Then

x, (x) = is an almost-periodic function of x for PROOF.

I'

(x+/y)

I(x + Iy)

Iyl ;> H

= sup (1m tp(k».

It follows from the representation (9) off(z) that co

x, (x) =

P.

v. ~ z~ak

•

-00

If O"k denotes the zero corresponding to then

OJ;

after an E-translation

T,

so that

Ialii +'t- ak I < a, (10) 00

< Mi•• ~ I z...!ckra. -00

• We may assume, with no loss of generality, that the set

{ak}

does not contain o.

Sec. 2] Here

CHARACTERIZATION OF A SET OF ZEROS

M~

is a constant such that, for

Iz-a" I< z - ck

M

a

Iyl :> H + 6 (6

and

451

> 0),

I

z - ck z-aflt -",

I<

Ma·

The sum on the right in (10) is periodic and therefore bounded. The lemma has been proved. LEMMA

3.

If ("-+-k=O. 1.2•... ).

a/c=ck+Hk) where tp(k) is a bounded function and

co

f(z)= P. V

II (1- :J,

(9)

-CX)

then in order that the function

O(x+la)=ln{f(x+la)ti~~}

(a>O)

be bounded for all sufficiently large, fixed values of a. it is nec(!ssary and sufficient that there exist a constant M such that for any h :> 0 and arbitrary integral T CX)

1~[<jI(k+'t)-<jI(k») k~~h31<M.

(11)

-CX)

where M is independent of T. PROOF.

From (9) we obtain

f(;t~~t) tift

6 (x+c't+la)-6 (x+la)=ln N

=

lim {In

IT Ck+~(k~-c"'-x-Ia +/'It't} ck+ljI(k)-x-ia -N

N-tocx)

II ck+ -x-Ia ck+o/(k)-x-Ia N-~

-

· In I 1m

N-tocx)

oJ-

(k+~)

-N

~

+ N!'If co {I n IIC(-N-q)+~(-N-q+'c)-X-Ia+I1f't} c (N-"'+q)+IlJ(N-"'+q)-x-la • f- 1

'

Since every factor in the product in the second term tends to e-i ", the logarithm in this term tends to -i1TT and so the whole term tends to zero. Hence, fJ (x+c..+la)-6(x+la) CX)

~ [ = P. V. """" In 1

I} (k + "') - ~ (k) ] + ck+':' (k)-x-III •

(12)

-CX)

The boundedness of 9(x + ia) in - C() < x < C() is obviously equivalent to the uniform boundedness of 6(x + CT + ia) - 6(x + ia) for 0 < x < c and for all

452

ALMOST-PERIODIC FUNCTIONS WITH BOUNDED SPECTRA

[APPENDIX VI

integral values of T. We will show that the uniform bounded ness of the righthand side of (12) is equivalent to the condition (11). We first estimate the size of o/(k+")-~(k) 1 1 ck+ Hk)-x-ta •

== SUPt ltp(k)1 and

We write M

'0=-co sup It-Ial < ,< col t-tal-M = for a

> M.

a

a-M

We then have

0/ (k + ,,) - 0/ (k) 1 ~

I ck

+ 'Hk) -

x - ta

2M/II ...... I ck - x -ta I '

and for sufficiently large a this expression can be made less than t. Using the inequality \In (1 - a) - a I < Ia 12 , which is valid for 1«1 < t, we find that the uniform boundedness of 9(x + CT + ia) - 9(x + ia) for 0 <: x <: C and for all integral T is equivalent to the boundedness of 00

~ 0/(k+")-4(k) ~ ck+'\I(k)-x-ta

(13)

-00

for all integral

I ~ 00

~

-00

T.

Actually this equivalence can be obtained directly by noting that

+,,) - I} (k) 12 < (2MI ck + IjI (k) - x -ta ~ (k

\9 aJ

00

~

~ I ck -

1

x -ta 18

-00

< ka

(O~x~c),

where klJ is a constant. From a comparison of the expression (13) with 00

~ IjI (k

~

+ ")-4 (k) ck-ta

(14)

'

we obtain

-co

-00

where K' and K" are constants. Finally, a comparison of (14) with the left side of (11), and the use of the notation h == a/c, yields

I

00

00

00

~ojI(k+")-4(k)_ ~~(k+'t)-cjl(k)kl<2M~ k ~ k-~ ~ ~+~ ~~+~ -00

-00

-C'X)

Therefore the bounded ness of 9(x + ;a) for all sufficiently large, fixed values of a > 0 is equivalent to condition (11). The lemma is proved.

Sec. 2] (a

453

CHARACTERIZATION OF A SET OF ZEROS

It is useful to note further that if 8(x + ia) is bounded for any value of a > sup" 1p(k», then it is bounded for all larger values of a. 4a

We can now give the following complete characterization of the set of zeros of an almost-periodic function belonging to the class [~]. THEOREM 4. For fez) to be an almost-periodic function of the class necessary and sufficient that the follOWing conditions hold: (a) fez) can be expressed in the form

[~],

it is

N

II (1 -~) a"

I(z) = c Um

(c a constant);

(15)

N-+oo -N

(b) the zeros at off(z)form an almost-periodic point set and can be given by the formula 'It

ak =~k+Y(k).

where 1p(k) is a complex-valued, bounded function; (c)lor any integral T we have the inequality le.. [o/]I

e. [0/] =

00

~ [o/(k

+

< M, where k

T) -

o/(k)]

-00

k

2

+h

2

and M is independent oj T. PROOF. NECESSITY. If fez) belongs to the class [~], then from Lemma 1 the set {ak} of its zeros is an almost-periodic point set. Moreover, from the corollary to Theorem 2 we have ak = k+Hk). (6) where 1p(k) is a bounded function. Thus condition (b) is satisfied. It follows from (7) that the product

i

N

11(1 -~) N-+oo -N a"

!p(z)= 11m

converges and that q:{z) is of completely regular growth, while the indicator diagram of q:{z) is the segment [-i~, i~]. The quotient J(z): 'P (z) ... By a slight modification of the argument used to prove Lemma 3, one can prove that if ak .. ck + tKk), where 1{I(k) is bounded, then the condition If"IRe 1/t(k)1I c; L < co is necessary and sufficient for an entire functionj{z) of the form (9) to satisfy 0< ma .. If(x

+ ia)1

C;

Ma < co,

where m" and Ma are constants and lal > sup 1m tKk). Heific [1) has remarked that the (. > 0) is necessary and sufficient for the condition If"IRe 1/t(k)]1 C; cr 1n(1 + I'l'l) + 0(1) inequalities ma(1 + Ixl)-· c; If(x + ia)1 C; Ma(1 + Ixl)'. Finally, Heific has shown that the condition 00

,,-~oo

exp{ ~f"IRe1/t)} < co

is necessary and sufficient for j{x) to belong to L'( - co, co) (p

> 0).

454

ALMOST-PERIODIC FUNCTIONS WITH BOUNDED SPECTRA

[APPENDIX

VI

is an entire function of exponential type with no zeros, and its indicator diagram consists of a single point, the origin. Thus the quotient.f(z): H (15 > 0) the function In /.f(x + ;a)1 is bounded. Moreover, argf(x + ;a) = + x(x) where X(x) is almost-periodic (Chapter VI). Thus for lal > H the function

-ax

(j

(x

+ ta) =

In I(x

+ ta) ei4:i:

is bounded, and so from Lemma 3 it follows that condition (c) holds. SUmCIENCY. Let .f(z) be expressed as in (15) with at == ('IT/a)k + 1p{k). Thenf(z) is an entire function of exponential type a. It also follows from Lemma 2 that

I' (x + iy) I{x

+ iy)

+

.

Ifl.

is an almost-periodic function of x for Iyl > sUP_co
+ ia) ei4

is almost-periodic for all sufficiently large

f(x

+ ia).

:l1

lal. The same result holds for

From the Phragmen-Lindelof theorem for a strip, fez) is bounded in any strip parallel to the real axis, and since it is almost-periodic on any straight line, it is almost-periodic in the whole plane. Its whole spectrum is in the segment [-ia, ;a]. Since all its zeros are in a strip, from Corollary 2, Theorem 2, § 2, Chapter VI, it follows thatf(z) belongs to the class [fl.]. The theorem has thus been proved. REMARK.

If the zeros at can be given by the formula

a,,= ~k+~(k), where 1p{k) is almost-periodic, then for any e > 0 there is a 15 > 0 such that every I5-translation of 1p{k) becomes, after multiplication by 'IT/a, an e-translation of .f(z). Hence the module of fez) belongs to the module obtained when the smallest module 9Jl('P, 'IT) containing the number 'IT and the module of tp{k) is multiplied bya/'IT. I

See B. M. Levitan (IJ.

Sec. 2]

CHARACTERIZATION OF A SET OF ZEROS

455

When this result is compared with Theorem 2, we find that when the conditions of this theorem are fulfilled the module of f(z) is the same as the module IDl(tp, 1T) multiplied by tl/1T. We now prove that condition (c) is independent of the remaining conditions of Theorem 4. In addition to this, we prove that there is an almost-periodic function tp(k) for which the set of functionals L~[tp] (±'T = 0, 1,2, ... ) is not bounded. The set of aU almost-periodic functions forms a complete, normed space with norm lI'fll sup II\I(k) I·

=

-CD

< Ii < 00

The functionals L~[tpJ are linear in this space. By a well-known theorem, a set of linear functionals, bounded for every element of the space in which they are defined, is bounded in norm, i.e., if for all almost-periodic functions we have (± 't = 0, 1, 2••...).

then there is a number M such that

IL, ['fJl < MII~II The functional

L~[tp]

(±'t=O, 1, 2, ... ).

(16)

can be written in the form 00

L,

['f! =

~ ~(k) [(k~~~ hI

kll~ha ].

-00

The kernel (k - 'T)/[(k - 'T). + h 2J - k/(kl + hi) changes sign at two points kl and k., where k. - 'T and kl - 0 for 'T - 00. For fixed 'T, we set tp(k) = -1 (k1 < k < kJ. We also choose the number T > 0 so that k. - kl + T will be an integer and so that - T+k

00

( ~ +T+t, ~)I (k!~~hll -- t3~ha I< I, -00

and define tp(k) = 1 on the partial interval kl < k <; T tp(k) to the whole axis by periodicity, using the period We then have

+ k.. (JJ

We then continue = k. - kl + T.

CD

L~['fl

~I

>~

k-"

(k-,,)lI+hll

ka~hlll·-1 =2In't+O(I).

. . 00

which contradicts (16). Thus the set of linear functionals L~[tp] is not bounded, and so there is an almost-periodic function ~k) for which the family of functionals is unbounded.' Condition (c) is difficult to verify, god so it is useful to have the foUowing sufficient condition that it be satisfied. • Sucb a fuoc:tion can in

fad be c:onstructed.

456

ALMOST-PERIODIC FUNCTIONS WITH BOUNDED SPECTRA

[APPENDIX VI

THEOREM 5. If Ihe Fourier series of an olmos I-periodic function tp(k) is absolulely convergenl, Ihen I L~ ('~II M, i.e., condition (c) holds.

<

PROOF.

We write co

t')(~) = 4[ eiAx ] = 2i (e ib -

~ !:as~ ~~

I)

or, otherwise,

k=l

fl (A) =

l

ItShh(K-A.)(en~_I) sh wh

(O
_ uh ~(::A.) (eib_l)

(-1t ~ A < 0).

Thus ,?(A + 211') = ,?(A).6a Since '1' ..(/) tends uniformly to tp(/). then for fixed ~

Hence if

T

(17)

we have

(1\1,,1-+ L (1\11· t

co

1\1 (k) = ~ cJ'ili" -CII

then co

Lc (~l or

= -co ~ ci) ()'J) co

IL (\jill ~ 2 I

t

(

~

~ I cJ I

)

It sh hit sh h •

-co

The theorem has thus been proved. Condition (c) can be transformed into another form. To perform this transformation, we prove an interpolation theorem that is interesting in itself. THEOREM 6. In order that there exist an entire function of exponential type not exceeding 11', bounded on Ihe real axis, and laking Ihe values (-I)ktp(k) for all inlegral values k of the argument where tp(k) is a bounded function, il is necessary and sufficient Ihat for all integral t'alues of T we have

ILcII\lII<M. where M is a conslant not depending on PROOF.

(II)

T.

We form the interpolation series co

&(z) = SI°ltftZ

{~' Hk)[z

1

11+ !J+ 4!O)}.

-co

This series converges uniformly in every finite region to an entire function of exponential type not greater than 11'. We have the relation (J(k) = (-l)"tp(k). 611 Note that in this derivation we do Dot assume that .,. is an in.....

Sec. 2]

457

CHARACTERIZATION OF A SET OF ZEROS

We will prove that when condition (II) holds the value of I,,<x) I (- 00 is bounded. Let z = x ± ih, Ixl I and let T be an integer. Then

<

{) ( -+-lh+ ) _ fl ( -+-lh) x_ or X._

= sin

< x < 00)

co

11:

(x ± II&) ~ +
If condition (II) is satisfied, the absolute value of the sum on the right is bounded for all T, arid so (Ixl~ I).

Ifl(x±lh+or)-f)(x±lh>l<M1

where Ml is a constant. Thus ,,<x ± ;h) is bounded for - 00 < x < 00, and from the Phragmen-Lindelof theorem ,,<x) is bounded on the real axis. Conversely, let it be given that there exists an entire function 1?(z) of exponential type not greater than 77, bounded on the axis, and such that 1?(k) = (-lYlP
where M is independent of T. We consider the function co

l(Z) = SI:1I:Z

{~~) + ~' ~(k) [z

1k

+ !]}.

-ex>

This is an entire function of exponential type, with an indicator diagram in the segment [-;77, ;77]. The function 'Y..

( z)

=

&(Z)-T(Z) sin KZ

is entire and of exponential type, and its indicator diagram consists of the origin only. We will obtain information on the size of this function on the imaginary axis. From the definition of y(z), we have 00

I1 (ly) I ~ ~ I sh r.yl {m+ 2 ~ k I~~I'YI} 1

1

00

~~ Ishr.yl {I~I + ~ Iyl (~kt~yIIY} 1

< ~ Ish 'lty I {_I + 1I:YKTYi} Iyl va· 11:

Moreover, the bounded ness of t?(z) on the real axis yields

I fl (z) 1< Ne" 1.1. ~t follows from the last two inequalities that x(6') = O(ViYf), and since x(z) of type zero, we have x(z) = const. We thus have

IS

~(z) =

co

slnK 1I:z {c+

9~O) + ~' ~(k)[z -ex>

1 k_

!]}.

458

[APPENDIX VI

ALMOST-PERIODIC FUNCTIONS WITH BOUNDED SPECTRA

For

T

even, we thus have CD

o(lh+'i:)-n(lh) = S~"d ~1\'(k)['h_~+":_1I11

k]

-CD CD

=shtth ~ Hk+..:)-Hk) Itt ~ Ih-It ' -00

and from the boundednes5 of O(ih

+ x) we obtain

CD

I~ [I\'

(k

+ or) -

~ (k») kl ~ h'

-CD

I'"

Sh"" h I f} (lh+or)-& (ih) I

h

CD

+ 2M ~ -00

where C is a constant. When

T

2

k +h

2

< CM,

is odd, we have

CD

L" ['tl

= ~[Hk+'t-

1)-9(k») kll!h a

-CD

00

+ ~~ (k+'i:-l)[(k~~~hll- kI~hi]. -CD

and 50

(M1 a constant ). The theorem has been proved. When we compare this theorem with Lemma 3, we obtain the foUowing theorem.

= + 9(k)

THEOREM 7. Let a" ck is bounded. For the function

(-+- k

= O.

I. 2 •... ). where tp(k)

CD

I(z)= P. V.

IT (1 - a:) -co

to satisfy the inequality

Iln[j(x + ia)enrx/c] I <; Ma 6b on every straight line y = a(lal > SUPk ",(k», where Ma is a constant, it is necessary and sufficient that there exist an en/ire junction of exponential type not exceeding 'IT, bounded on the real axis, that takes the values (-lfflk) for integral k. Theorem 7 can be used as follows in the reformulation of Theorem 4.

6b It can be shown that this condition is equivalent to having the entire functionj{z) be of sine type; that is,

0< mIl

'" Ij{x

+

iY)le-wl)'1 '" M"

< 00;

Iyl

;> II.

Sec. 3]

THE RELATION BETWEEN THE FOURIER SERIES OF

tp(x)

AND

f(x)

459

THEOREM 8. For f(z) to be a function of the class [11], it is necessary and sufficient that the foliowing conditions be satisfied: (a) f(z) can be expressed in the form

N

/ (z) == lim c

II (I - : );

N-+ot>

-N

•

(b) the zeros akform an almost-periodic point set; (c) the zeros at can be given by the formula

a" = V'(z)e-iwz

where real axis.

;

k+'f(k).

is an entire function of exponential.type H <;

fT,

bounded on the

3. The RelatioD Between the Fourier Series ortp(x) aDd fix) THEOREM 9. Let tp(x) be any almost-periodic function with absolutely convergent Fourier series, and let

a" =

1t

T k

+ '1
(::±: k

=

0, I. 2, ... ; 11

> 0).

Then f(z), defined by (IS), is a function of the class [11] with an absolutely convergent Fourier series. PROOF. We start from the relation (12). Setting x = 0 and replacing by x/c in this relation, we obtain

In

I(x+la)ei~a: I(la)

=

~

[

;(k+:)4
P. V. ~ In 1+ ck+cjI(k)-la

T

r.)

c=X.

lc- -IXI

for x

= cn (± n = 0, In

1, 2, ... ). For sufficiently large

lal.

l(x+la)eiAa: _ ~ 'l-(k+: )-4'(k) I (/a) - P. V. ~ ck+~(k)-Ia lc .. -IXI

~(_1)"-1 ~

+ ,,-I ~

n

~ lc_-IXI

(,(k+:)-Hk»)" ck+Hk)-la '

(18)

for all real values of x. The hypothesis of the theorem yields IXI

~(x)= ~ CtrUfD

(19)

-IXI

If we use

11'1'11

GO

=

Ilc,l

for the norm in the ring (W) of functions of the

-GO

form (19), then for any fixed k we have (20)

In addition to this, for n

:> 2; we have

460

ALMOST-PERIODIC FUNCTIONS WITH BOUNDED SPECTRA co

~ (Ck+'JI(~)-la)n

-co

[APPENDIX VI

=OCa:n-

J ).

Thus for 101 > 2 IIY'II the second series in (18) converges in norm, and so it converges to an almost-periodic function with an absolutely convergent Fourier series. The first sum in (18) can be expressed in the form co

P. v. ~

ljI(k+~)-
co

P. v. ~[Y(k + :)-y(k)]

=

ck+,}c(k)_la

-00

-00

ck

.

X cllk 3 +all

~

<}(k+ :)-
+ ta ~

-co

co

- ~[y(k+ :)-'Hk)] (Ck+Hk):!~~)(Ck-la)

•

-00

The convergence in norm of the second and third ~eries follows from (20). From (17) we infer that the first series is also convergent in norm. Therefore the function 00

[

O(x) = P. v. ~ In 1 + -00

1/1( k + ~ ) - 1/1 ( k) k (k) _ . e + 1/1 za

1

(e='fT/a)

is almost periodic with an absolutely convergent Fourier series, and all the frequencies in this series belong to the numerical module e - 101t1jl or (~/ 'fT)0lt1jl. The same statement applies to the function 00

x(x)

= e'(x) =

~ ekei""x; -00

that is, ~~oolekl < 00 and I-'k E (~/'fT)0ltIjl' If we consider the function X(x) only at points of the form x = en (± n = 0, 1,2, ... ), the frequencies I-'k can be replaced by numbers that are congruent to I-'k modulo 2'fT / e = 2~ and lie in the interval [0, 2~]. We obtain an expansion 00

XI(X)

=

~ dkeiY,X,

where ~~ooldkl < 00 and 0 <;; 'Yk <;; 2~. In addition, X\(x) coincides with X(x) at the points cn, and (because of (18» with the function f-l(ia)j(x

+

ia)e ifu .

The difference f(x + ia) - f(ia)XI(x)e- ifu is clearly an entire function of exponential type <;; ~, bounded on the real axis, and vanishing at the points cn. Hence f(x + ia) =f(ia)x\(x)e- ifu + c sin ~x,

Sec. 3J THE RELATION BETWEEN THE FOURIER SERIES OF tp(x) AND f(x)

461

where c is a constant. Consequently f(x) is represented by the absolutely convergent Fourier series 00

f(x) = ~ Ckel>."", -00

where the exponents ~ belong to the smallest numerical module that contains (~/ '1T)~ and the number ~. The example given at the end of § 1 shows that the converse of this theorem does not hold. In view of this fact, the following theorem is of interest. THEOREM 10. If the Fourier series of afunctionf(z) of the class [a] converges absolutely, and if the sequence {ak}~CIO of its zeros satisfies the condition

, Re ak - !2.. k . :1

s

I< ..:...4:1'

where s is any real number, then ak = (1T/a)k + tp(k), where tp(k) is an almostperiodic function with absolutely convergent Fourier series. PROOF. We prove this theorem by using the general theory of rings, as described by I. M. Gel/fand in [1]. We consider the ring of almost-periodic functions tp(k) defined on the set of integers ±k = 0, 1,2, ... with absolutely convergent Fourier series, and take for the norm the sum of the absolute values of the Fourier-series coefficients.' The set of values of the functions 0). For z E I we have If(z + ('1T/a)k + s)1 ;> m > 0 for all integral k, and so f(z + (1T/a)k + s) is not in any maximal ideal of the ring. Hence, for z E I, the function

f'(z+i k+S) f(Z+~ k+S) , belongs to the ring and is continuous in norm. Since the operations of summation and taking a limit do not lead out of the ring, the function

~(k)= 2!'

I( f'(Z+~k+S) : ) f (z+jfk+s

zdz-s

belongs to the ring. In other words tp(k) can be expanded in an absolutely convergent Fourier series. The following theorem follows easily from the last two theorems. , To avoid ambiguity in the expansion of such a function in a Fourier series, we may USUme that its spectrum is in the half interval (-11',11'].

462

ALMOST-PERIODIC FUNCTIONS WITH BOUNDED SPECTRA

[APPENDIX VI

For the sequence {at}~"" of zeros of a functionf(z) of the class [~] to be expressible in the form Ok = k+~(k), with 1p{k) almost-periodic and expressible as an absolutely convergent Fourier series, it is necessary and sufficient that, for any positive integer p exceeding some Po, all divisors off(z) of the form THEOREM

II.

i

co

,1',l..z)= •

p.v.n(I-~) -co 1'k+r

(r=O. l. 2•...• p-I)

can be expanded in absolutely convergent Fourier series. PROOF. If 1p{k) can be expanded in an absolutely convergent Fourier series, then so can 1p{pk + r), and from Theorem 9 so canfp.,(z). Conversely, let all the functions jp/z) be expansible in absolutely convergent Fourier series for all p greater than some Po. If P is chosen so that the inequality l1p{k)1 < 1fP/4~ is satisfied, then

IRe

O1'k+r -

?k -

':

I< ;~ .

From Theorem 10, taken together with this equality and the absolute convergence of the Fourier series of fl1.'(Z), it follows that apt+, can be expressed in the form o1'k+r .... k+~1'.,(k) (r=O. 1.2•...• p-I), where tpp.,(k) is an almost-periodic function, expansible in an absolutely convergent Fourier series. Setting

'It:

QJ (z) =

(n n

sin ;

(Z_r»)2

I _ _ __ _f"...!.:I.1. f":li

stn..!:..(j-r) p

and defining 1p{k) by the absolutely convergent series 1'-1

~(k) = ~ QJ(k)[~p.J( k P j)- ~ ], 1-0

we obtain the representation Ok = : k +~(k). The theorem has thus been proved. We have shown that if the zeros of a function of class [~] can be expressed in the form (6) and jf 1p{k) can be expanded in an absolutely convergent series, then the corresponding function f{z) can be expended in an absolutely convergent series and j(z) has an infinite set of divisors of the same type.s

• v. P. Potapov in [I], using results obtained by M. G. Krein and the author in [I]. showed that every almost-periodic polynomial is infinitely divisible in the ring of entire. almostperiodic functions of the class [~] with absolutely convergent Fourier se~. and ga.ve a complete classification of all such divisors. We mention that. as was shown m Append~ ~, almost-periodic polynomials are not necessarily divisible in the ring of almost-penodlC polynomials.

Sec. 3]

THE RELATION BETWEEN THE FOURIER SERIES OF tp(X) AND

lex)

463

In Appendix V we considered a special factorization of an almost-periodic function/(x) with a bounded spectrum. We showed that, if/(z) can be expanded in an absolutely convergent Fourier series and f(x) ;> 0 (-ex> < x < ex», then I(x)

= Icp (x) Ill.

where cp(z) is almost-periodic, has a spectrum half the length of that of lex), and has an absolutely convergent Fourier series, and the zeros of cp(z) are in one half-plane. We can immediately show that the requirement in this theorem that the Fourier series of f(x) be absolutely convergent is essential. In fact, we can construct an example of an almost-periodic function of class [d), expressible in the form

p.V·II (1- ,,:)(1 -·iJ 0)

I(z) =

(1m air

> 0)

-0)

and having as a divisor the function GIl

cp(z)= P. V.

II (1 -

~),

-0)

which is not almost-periodic. To obtain this function we set arc = k+ lljl(k),

where tp(k) is a positive, aJmost-periodic function not satisfying condition (c) of Theorem 4. From this theorem, it follows that cp(z) is not almost-periodic. We number all the zeros of/(z) so that the zeros a" have even numbers and the ti" odd numbers and form the sum 0)

L~[;)= ~[IjI(P+"t)-;(P)1 pI~hl. -0)

For even p, the quantity fj1{p tp(p + 1) and so

+ 'T) -

fj1{p) differs in sign from fj1{p

+ 1 + T) -

0)

~ [;) = ~ [;(2k +'t)-H2k») [(2k~+ h' -0)

-

(2k!klt~ hI

J.

Hence

ILr [IJI) I < cit sup IIJI (k) I.

rc -here Cia is a constant independent of T and tp. Thus f(z) is an almost-periodic function of the class [d), is positive on the real axis, and can be written as/(z) = cp(z)cji(z) with cp(z) not almost-periodic.

ApPENDIX

VII

MISCELLANEOUS THEOREMS AND PROBLEMS 1. Let fez) be a holomorphic function of proximate order p(r) inside the angle larg zl __ ex (0 < ex __ 7T/2p) and let h,(±ex) = h,(O) cos pex. Then h,(~)

for all values of () satisfying ( - ex

=

h,(O) cos pO

< () ~ ex).

PROOF. Writing qJ(z) = f(zI/P), we obtain a function of order unity, holomorphic in the right half-plane, such that the relation

h,(-+- ( 1) = h, (0) cos a 1 (41 = pal is satisned. We find that the indicator diagram of qJ(z) lies inside the angle formed by the rays arg [z - h,(O)] = (7T/2 + exI), and the point h,(O) belongs to the indicator diagram. The above assertion follows directly from this. 2. Let fez) be an entire function of order p(r) (p < I), let all the zeros off(z) he on the negatil'e real axis, and let 'nf(r)~ _11:_ rP (r). sin 11:p Then n(r)~rp(r) (I'itchmarsh). PROOF.

It follows from the equality co

II n f(z)/ =

If

In

(I +7) dn(t)/

o

00

=/n(t)ln(' and the inequality nCr)

+f) oI+ ft~~t!)dfl

< crP(r) that

f

on

/Inf(z)/-
tp (t/-l

It+Zl

dt ,

o

and that for larg zl __

7T -

"

(6

> 0) we have

f

00

Ilnf(z)1

<

clr PI ,.,

o 464

tP(t)-ldt

It-,"I

< carP

(r),

APPENDIX VII]

MISCELLANEOUS THEOREMS AND PROBLEMS

465

where C6 is a constant. Let !p{z) be an entire function with zeros on the negative axis such that n..(r) ~ rPirI. Then (Chapter I, Theorem 25) In cp (z) ~ J (1t - 6) rP (r)

(z =

Te CS).

The function , (z) ~

= In/(z) In, (z)

< ." -

is holomorphic and bounded in the angle larg zl 6, and it tends to unity on the negative real axis. It thus follows that tp(reiO) - I on any ray in this angle. Thusf(z) has completely regular growth and so nCr) ~ r"Crl.

3. If f(z) is holomorphic and of order p inside and on the boundary of the angle 101 .;;; Cl, 0 = arg z, .,,/2p < Cl < .,,1p, and if ht<±Cl) ht
<

function of order one inside the angle larg zl .;; Cll (Cll = Pcl, .,,/2 < Cll < .,,) and h,.(±Cll) h.,(O) cos Cll' On the other hand, the fundamental relation for the indicator (Chapter I, § 16) yields h,.(Cll) + h,.( - c lI) :> 2h..(0) cos Cll' and so h..(±Cll) = h.,(O) cos cll • The three supporting lines x = h,,(O), x cos Cll + , y sin Cll - h..(cll ) = 0, and x cos Cll - Y sin Cll - h( - at,) = 0 intersect at z h..(O), and thus the whole indicator diagram of 9'
<

<

<

<

<

perhaps a set of zero density), and Ii II (r) r ..moo r,lr)

sin 1I:p h (0)

= -,,-' . PROOF. For cl = ." we have .,,/2p < cl < ""!;. and so f(z) is of completely regular growth for 101 < ." and ht
A=

II:

trp ",(0).

466

MISCELLANEOUS THEOREMS AND PROBLEMS

[APPENDIX VII

5. If p < t, h,{-rr) < h,(O) cos '"p, and the function fez) is of completely regular growth along any ray arg z = 60 ~ '", thenf(z) is of completely regular growth in the whole plane, ~,(¥') has only one discontinuity at '", and the density of the zeros is (*)

Conversely, if the density of the set of zeros exists and ~(¥') has only one discontinuity at '", then the density can be obtained from (*) and h,('") = h,(O) cos '"p. The second assertion follows from Theorem 25, Chapter I, the first from Theorem 6, Chapter III. 6. If an entire functionf(z) is of completely regular growth and of integral proximate order per), then 2.

! f h (6) e ed6. ip

0, =

o

For the proof we use the formula I)

h(6)=

f 'Psinp'Pd.:1(cp+6)+8,cosp6 -2ft

(Chapter II, § I) and the identity o

fe

4f• dA. ('P)

= O.

7. Every trigonometrically convex function h(6) with exponent p can be expressed in the form t

h(6) = h (60) cosp (8 -60>

+ h' p(80) sin p (6 - 00)+ 2'1t ,f sinp ('P- O)ds ('P),(*)

where

s ('P) == ~

[It' ('P) +

pI

I•

h (,) dCP]

(A. Pfluger ).

In fact if s(1P) is a differentiable function, then h(fJ) is a solution of the equation

h" + p'lh = 2'1tps'. Writing the solution of this equation in terms of a Cauchy function (in § 16, Chapter I, Green's functions with periodic boundary conditions were used) we obtain (*). The general result can be obtained by approximating s(1P) by differentiable functions.

MISCELLANEOUS THEOREMS AND PROBLEMS

APPENDIX VII]

If we now differentiate (*) with respect to 6 and write Q(6) (i/p)h'(6), we also obtain

Q (6) e-ipe = Q (6 0) e-ip&" -

21t1

467

== h(6) -

f•

e-ip'f ds (cp).

80

8. Let f(z) and tp{z) be entire functions of exponential type T, and T", where h,( -1T/2) == h,(1T/2), h,,( -1T/2 + at) == h,,(1T/2 + at), and the integrals co

co

f

In ''(x) I d 1

+ Xli

, (xe'·) I d SIn I1+x

x,

3

X

-co

-co

exist. Then the exponential type of the function f(z) tp{z) is 1

't

= ('t~+ 't! + 2\ cos IX I'tr,r?:

(S. N. BernStein).

In this case the indicator diagrams of f(z) and tp(z) are segments. Both functions are of completely regular growth. When they are multiplied, their :, indicator diagrams are added. Thus the indicator diagram of the function 'I
= e-fa/(z)

is an entire function of exponential type, bounded on the real axis. Its indicator diagram is a segment of the imaginary axis or the point O. The first of these alternatives is impossible, since when functions are multiplied their indicator diagrams are added, and the indicator diagram of f(z) is a vertical segment passing through the point a. Thus the exponential type of f{z) would have to be greater than a. Therefore tp(z) is of exponential type zero, and since it is bounded on the real axis, we have tp(z) == const. 10. Let F(z) be an entire function of exponential type and let IF(x)I~M

(-00

< x < 00).

(*)

Let Q(z) be an entire function of exponential type, let all the zeros of Q(z) be ~ of F(z), let ho(O)

Den the function

== ho("')' and let lim IQ «(v) I e- fll " Iym > O. lI-+±co

., (z) = ., II polynomial

F(z) (z)

{!

of degree not greater than m (N. I. Ahiezer).

(**)

468

[APPENDIX VII

MISCELLANEOUS THEOREMS AND PROBLEMS

PROOF. The function
k, (6) = h,(6) - kg (6). We have h., (0)

+ h, (1t) = h,(O)+ h,(1t) -

(h g (0)

+h

g

(1t»).

The indicator diagram of F(z) is a segment of the imaginary axis, i.e., we have h~O) + h~1T) = 0, ho(O) + h n(1T) ;> 0 because of the properties of indicators (Chapter I, § 16); consequently

<0

h., (0)+ h, (1t)

or h, (0) == h, (1t) = O.

On the other hand it follows from (**) that

;)<0.

h,(~

Hence the indicator diagram of (Iy) I

IF (z) 1< Meo I" I,

< ely I'"

(c constant ).

The entire function of exponential type zero til-I

~ (z) == [ q> (z) - ~"

(Ie)

k/O)

z1c] z--

1e=0

is bounded on the imaginary axis, and so is a constant. Hence (Ie)

til

q> (z) ==

~

" k/O) zle.

1:-0

11. LeI d, be lhe length of the shortest interval containing all the points for which the function of a real variable fix) differs from zero. Then iffix) and
~(x)=

J j(x-t)q>(t)dt, -co

it follows that tJ. = d,

+ d" (Titchmarsh).

MISCELLANEOUS THEOREMS AND PROBLEMS

APPENDIX VII]

469

PROOF. It, follows from the Paley-Wiener theorem that the indicator diagram for the entire function CD

f ,Ooz/(x)dx

x,(A) =

-CD

is a segment of the imaginary axis of length d" and that the indicator diagram of CD

X,(A)=

J'~(x)dx -CD

is a segment of the imaginary axis of length fl.. The functions X,(A) and x.(A) are of completely regular growth. Therefore the indicator diagram of X. (A) = x,(A)X,(A) is the sum of those for X,(l) and X,,(l), and it is thus a segment of the imaginary axis of length d, + d". It follows from the representation CD

f

X4' (A)'=

'~Hx)dx

-00

and the Paley-Wiener theorem that the shortest interval outside of which ,ex) - 0 coincides with the conjugate diagram of X,,(A). Thus d" - d, + fl.. 12. Let f(x) and I}'(x) belong to the class L(O, ,,) and let 111

J1(/),(x-/)dt=O

o

in (0, ,,). Thenf(x) - 0 almost everywhere in (0, at),l}'(x) - 0 almost everywhere in (0, fJ), and at + (J - " (Titchmarsh). PROOF. The Laplace transforms T

F(z) =

T

f I(/),-'·dl,

~(z)= f,(/)e-'.dl

o 0 of the functions (assumed to be zero outside the range of integration) are entire functions of exponential type, bounded on the imaginary axis. They are of completely regular growth, and their indicators in the right half-plane are h,.(8) = -/I cosO, ",(6)= -~cos8. The indicator of the entire function of exponential type 2y

F (z) ~ (z) =

thus has the form

111

f ,-In {f 1(/), (x T

I) d/} dx

0

h,. (8)= -(/I +~)cos8 <-Tcos8.

Hence at + (J ;>". It also follows from the "delay theorem" that f(x) almost everywhere in (0, at) and I}'(x) = 0 almost everywhere in (0, fJ).

=0

APPENDIX

VIII

FURTHER DEVEWPMENT OF THE THEORY OF FUNCfIONS OF COMPLETELY REGULAR GROwru We present a brief survey of results from the further development of the theory of functions of completely regular growth. Some other results have been given in footnotes added to the relevant parts of the book; we do not repeat these here.

1.SubbanmKRdcfuncdoDS The theory of functions of completely regular growth was generalized by Azarin [4-7] to subharmonic functions on the n-dimensional space Rn. The order and type of sub harmonic functions are defined in the natural way: p

= lim In u(x) Ixl--+co

.

a

= lim

Inlxl '

Ixl--+co

u(x)

IxI P

'

where x denotes an n-vector (XI' ... , xn ) and Ixl = (xf + ... + X;)1/2. The indicator h(e) of a subharmonic function (where e is a unit vector), relative to a proximate order p(r), is defined by the equation

r1m

u(re) p(r). r A CO-set is defined in the n-dimensional case in a way similar to that used in the plane case, as a set of hyperspheres such that h(-) e =

r-+co

I

R

lim - - ~ r.n R-+co R n - I J

I

= o.

The function u( re) is said to be of completely regular growth if the limit · u(re) I1m - = h(-) e

r-+co

rP
exists uniformly in e under the hypothesis that the end of the vector re lies outside a CO-set. The role of the zeros is played by the masses p.(E) that appear in the Riesz representation of u(re) in an arbitrary region. Angular density is defined as a function of domains U on the unit sphere by the equation 470

Sec.

I]

SUB HARMONIC FUNCTIONS

471

= lim

/l( U)

R-+oo

where KR = {re: e E U, 0 <; r <; R}. When p is nonintegral the masses are said to be regularly distributed if the liInit exists for all domains U for which /l(aU) = 0. 1 For integral p we require additional hypotheses siInilar to condition (B') of Chapter II. Azarin showed that a subharmonic function is of completely regular growth if and only if the corresponding masses are regularly distributed. Moreover, using the notion of the regularized indicator

= lim

h(e) ,

h(el ),

sup

6-+0 le.-e')<6

he obtained the representation h(e) =

1-

Itl=1

~(e, f)

dmf

+ Y(e),

(I)

where Y(e) is a spherical function (which is absent for nonintegral p), I mf(u) = ~ /l( U), (n - 2)(p + n - 2)mes{I~1 = I} and ~(e,

f) is a fundamental solution of e~

+

p(p

+

n -

2)~

= 0,

on the unit sphere, where e is the spherical part of the Laplacian. Azarin also showed that the "regularized" indicator h(e) of an arbitrary subharmonic function of proximate order p(r) can be represented in the form (1), where mf{u) is a mass distribution on the unit sphere. Consequently the representation (1) is a generalization of the representations (1.80) and (1.82) of the indicator of an entire function. For regularly distributed masses this formula yields a generalization of (2.04) and (2.08). A very useful tool for investigating the possible behavior of entire functions • the following theorem of Azarin [1] on the asymptotic approximation of a aiven subharmonic function of finite order by the logarithm of the modulus of an entire function. To every function u(z) that is subharmonic in the plane Izl < 00 and oj proximate order p(r), there corresponds an entire Junction J(z) such that lu(z) - InIJ(z)1I = o(l"'(r»

, : IThis condition is to be understood in the sense that zero is the lower bound of the measures of

6e measurable subsets of the unit sphere that contain

au.

472

FUNCTIONS OF COMPLETELY REGULAIl GROWIH

[ APPENDIX VID

outside some CO-set. In particular, as Azarin showed, it is easy to construct a function u(z) that is subharmonic in the plane and for which the set of rays of completely regular growth is an arbitrary closed set. The approximation theorem implies the existence of a corresponding entire function. This is a converse of the corollary at the end of Chapter III, § 1.

1. FunctioDs of completely regular growth in an angle Govorov [1; 1] investigated completely regular growth for functions holomorphic in an angle. We may suppose without loss of generality that the angle is the half plane II = (0 " arg z " '1T), since the general case is reducible to this special case. It is necessary to define the order of a function that is holomorphic in II. Titchmarsh defined the order of a function f(z) that is holomorphic in II to be the greatest lower bound of numbers ., for which iI

lim lnlJ(re )1 = 0

(2)

rP

r-+QO

uniformly in fJ, 0 < fJ < '1T. Govorov introduced a definition of order that is more convenient for applications. Suppose that we have the asymptotic inequality sup lJ(reil)1 < exp r" 0<.< ...

for some JL > 0; he then defines the order of f(z) to be the greatest lower bound of numbers ., for which (2) is satisfied for 0 < fJ < '1T. Here uniformity in fJ is not required. Govorov established the nontrivial fact that these definitions are equivalent. We note that the order introduced in this way describes not only the growth of f(z) in the half plane II, but also the rapidity of its decrease. For entire functions this order coincides with the order defined in terms of MJ..r). Just as for entire functions, the indicator is defined by

In'A;iI)'.

h(fJ) = lim r-+QO

One can also define the indicator with respect to a proximate order. The whole investigation depends on the following representation of holomorphic functions of finite order p ;;> 0 in the half plane 1m z > 0:

J(z) = exp[i(llo + a1z + ... +a"z")] X

1 exp -; '"

x

II

J

QO

-QO

(

tz + 1 t +1

-2--

I - z/z

k

IZkl>1 I - Z/Zk

),,+1 da(t)

--- X

t- z

(

exp 2i z 1m -

I

Zk

II

z - Zk IZkl <: 1 z - zk

--_-

Z" I ), + -z2 1m - I + ... +-Imq zl 2 z:

(3)

Sec. 2]

FUNCIlONS IN AN ANGLE

473

where q = [p), ak are real constants, Zk are the zeros of f(z) in II, and a(t)., lim

y-++O

(t lnlJ(x

Jo

+ ~)I dx,

which is clearly of locally bounded variation. This representation plays the same role in the theory of functions of finite order in a half plane as Hadamard's theorem in the theory of entire functions. Govorov obtained a necessary and sufficient condition for a function f(z) of finite order and normal type in a half plane to be a function of completely regular growth. The theorems involve both the distribution of the zeros of the function and its behavior on the boundary. More precisely, they involve both the masses in the half plane 1m z > 0 that are involved in the representation of the subharmonic function lnl.f(z)1 as a potential, and also masses on the •. boundary. In place of the function n(r, f, 9), Govorov introduces the function ~

c(r, f, 9) =

sin 9k ,

<"k<" .<8k <8 I

where rk e i8 are the zeros of .f(z). To take account of the boundary behavior he considers the function ( ) = (t da(x)

Jo

'1' t

(4)

1 + Ixl·

.PinaUy, he introduces the function

"e

a(r 9) = { c(r, 0, 9) - '1'(r), , c(r, 0, 'IT) - '1'(r) + '1'(-r),

0 <: 9 < 'IT, 9 = 'IT.

}

(5)

say that the set of zeros of a function .f(z) of order p has an argumentboundary density if the limit

~(9)

=

lim a(r, 9) "-+00

,."

'eXists for aU 9, 0 <: 9 <: 'IT, except for an at most countable set that does not DOntain 0 or 'IT • . Govorov showed that, for nonintegral p, completely regular growth in the ~ half plane is equivalent to the existence of an argument-boundary density. '" necessary and sufficient condition for .f(z) to be of completely regular growth '" the closed half plane is that the argument-boundary density exists, is ..,inuous at 0 and 'IT, and satisfies the following condition: . The limit

lim

1

r-+:!: 00

,.,.

r" dIp( x ) Jo 1 + Ixl

(6)

474

FUNCTIONS OF COMPLETELY REGULAR GROWTH

[ APPENDIX VIII

exists, where cp(x) is the singular part oj a(x); that is, cp(x)

lim T(r) '-+00

=.!. P

rP

lim T(-r) '-+00

= a(x) -

=.!. P

rP

fox InIJ(t)1 dt;

lim InIJ(r)l. '-+00

rP

,

lim lnl J(-r) 1 r-+oo

rP

Here the indicator function is given by the formula hI,'}) =

~

("g(I/I, ,'})cl A(I/I),

(6')

sm 'lTP Jo

where A(I/I) is the argument-boundary density and

Si~ 1/1 [cos p(l,'} g(I/I,,'}) = {

1/11 - x) - cos p(,'}

.

2p sm p(,'} - 'IT), -2p sin p,'}, 1/1

+ 1/1 -

'IT)],

0<1/1 < 'IT,

1/1 = 0,

=

'IT.

Govorov also obtained a necessary and sufficient condition for the functionJ(z) to be of completely regular growth when the order is an integer. For the open half plane 0 < fJ < 'IT the condition is that the argument-boundary density exists and that the limit

- r

a-,~~

[I

~

If

sin pOn

PI .. ,.<,--;:--2'IT

l"lxl'"

If

InIJ(x)1 dx xp+ 1 -2'IT

dcp(x) ]

1 .. lxl .. ,xP+1

(7) exists and is finite. The latter condition is similar to condition (B) in Chapter II, § 1. In this case the indicator is given by the formula hlfJ) = (2a - ap)sin pO

+ 2 r(J .1/1 or, sin p( fJ Jo sm 'Y + 2 r'IT 1/1, J(J

sin 1o 1/1-.dA(I/I) sm 1/1

+ 2 cos pO

'IT

~

- 1/1) dA( 1/1)

'IT sin p( fJ - 1/1) dA( 1/1), sm 1/1

(8)

where A(I/I) is the argument-boundary density and a p is the constant that appears in the representation of J(z). Govorov also found a necessary and sufficient condition for completely regular growth in the closed half plane for functions of integral order, and gave a representation for the indicator.

Sec. 3]

475

REGULARITY OF MASS DISTRIBUTION

3. Regularity of the mass distribudon and excepdonal sets In Chapter II we considered entire functions with a regular set of zeros; that is, a regularly distributed set satisfying a supplementary condition of "thinness", namely Condition (C) or (C') of Chapter II, § 1. For this case we defined exceptional sets. For example, under Condition (C) the asymptotic equations (2.03) or (2.05) are satisfied outside the disks Iz - ani <; dlan ll -P(!a,,!)/2. In order to define an exceptional set in the general case of functions of completely regular growth, Kolomiiceva [1] sharpened Lemma 7 of Chapter II in the following way: Let {an} be an infinite set with a limit point at infinity, and let

F6(Z) = II(1 -

z/an ),

(9)

where the product extends over the factors in which Ian - zl <; 8r; here Izl and 0 < 8 < Then for z =1= an

i.

\lln IF6 (z)llIn addition, let an e > 0 there exist 8

fo 6T(n (t)/t) dtl < In(l + 1/8)n (8r). z

z

=r (10)

~-set

>

be a set of points z in the plane such that for every 0 and r.,6 > 0 such that

lo

hr

nz(t) P( ) --dt<er r t

(11)

< 8 and Izl > r.,6' Then a necessary and sufficient condition for the asymptotic equality Inlf(z)1 R:: H(fJ)rP(r) on an infinite set is that the set is an

for h

~-set.

Returning to the "thinness" conditions (C) and (C'), we may remark that the exceptional set can be more restricted than in Theorem 5 of Chapter II. Suppose, for example, that (C) holds. Then the asymptotic equation (2.05') is satisfied on the boundary of each CR-disk Iz - ani = dlanll -P(!a,,!)/2, and Inlf(z)1 = Inl

/~~II I+ lnlz -

ani·

Applying the maximum and minimum principles to the first term on the right-hand side of this equation, and using (2.05'), we find that the asymptotic equation lnlf( re i8 )1

R::

H( fJ)rP
+ lnlz

- ani

holds throughout the CR-disk. If we delete the disks Iz - ani <; CP(lanl), where -In Ip(r) = o(rP(r», the asymptotic equation (2.05) is valid in the rest of the plane. Consequently ··uniformity" of the distribution of the zeros (or of masses in the case of

476

FUNCflONS OF COMPLETELY REGULAR GROWTH

[ APPENDIX YIU

subharmonic functions) leads to narrowing the exceptional set outside which the asymptotic equations (2.03) and (2.05) hold. We also note the following result of Azarin [11): Let denote a set of disks Iz - z.1 < r. satisfying the condition

C:

lim R-+«J

R-"{ Iz.1~ r;"} = 0
and let C~ denote a set that can be covered by sets C: for all p. > o. Then if the asymptotic formula (2.03) or (2.05) holds outside a cf set, it also holds outside a C~ set. Grisin [2-4) studied the following general problem: let u(z) be subharmonic in the whole complex plane (or in II+ = {z: 1m z > O}), and have formal order p(r); that is, for some constant M

sup u(z) <; M,.t'<'). Izl<'

Corresponding to this u(z) we construct the function

+ hz) + u(z)J. (12) A set C is called "exceptional" if u,,(z) :::t 0 outside this set as h --+ O. This u,,(z) = r--P(') [ u(z

means that to every £ > 0 there corresponds a 6(£) such that lu,,(z)1 < £ for Ihl < 6(£), z E C and z + zh E C. In this case if the masses p.(E) in the Riesz representation of the subharmonic function u(z) are regularly distributed, the asymptotic formulas (2.03) and (2.05), which hold outside some CO set, are easily seen to hold outside an exceptional set C. The regularity of growth of the subharmonic function is evidently greater, the narrower the excluded set is. Grisin distinguishes classes a), b), c) and d) of subharmonic functions with the formal proximate order p(r), according as the exceptional set C can be covered by a collection C of disks respectively satisfying the conditions: 2 a) C has arbitrarily small upper linear density, p.( C) < £. b) C has linear density zero. c) C subtends a finite angle at the origin; that is, «J

r.

I

Zj

~_IJI

< 00,

where 1j is the radius of the excluded disk of indexj, and Zj is the distance of its center from the origin.

ZWe atate Griiin'a results for func:tioDa that are aubharmoDic: in the whole pJane.

Sec. 3]

477

REGULARITY OF MASS DISTRIBUTION

d) Chas a finite sum of radii, ~flj < 00. Evidently each class is contained in its predecessor. If the set of zeros of an entire function f(z) is an R-set for the proximate order p(r), then, as we have just seen, the subharmonic function u(z) = Inlf(z)1 belongs to class d). In Hayman's theorem3 (as sharpened by Azarin) the subharmonic function in the half plane 1m z > 0 evidently belongs to class c). The following functions are used to describe the regularity of a mass distribution: 1. The density function C)(a) is defined by

~(a)

= lim sup p.( C(z: ar» , r-+ «J

,.,.< )

II

z = re il ,

0 <: a <: I,

(13)

where C(z, ar) = {f: If - zl < ar}. If C)(a) tends slowly to zero as a -+ 0, there is a sequence of points z" -+ 00 in whose neighborhoods most of the mass is concentrated; that is, in this case the mass distribution is not regular. 2. In the following approximation the regularity of the mass distribution is described by the function eB(r, a) =

s~p

p.( c~:)ar» - B( a),

(14)

where B(a) is an arbitrary nondecreasing function which is continuous on the right and satisfies the condition ~(a + 0) < B(a - 0). The function e;(r, a), as it turns out, tends to zero as r -+ 00, uniformly with respect to a (0 < a < I). Hence the function e(r) = sup

sup

e;(t, a)

(IS)

t>r 0<11<1

is monotonic and tends to zero as r -+ 00 (we omit the subscript B to simplify the typography). We can now state the fundamental theorem: THEOREM.

Let u(z) be subharmonic in the whole plane and of normal order

p(r). Then: 1. u(z) belongs to class a). 2. If the densily junction of the Riesz mosses satisfies

foln(l/a)d~(a) <

00,

then u(z) belongs to class (b). If (16) is satisfied and in addition the integral

'see footDotc II in Chapter V, f3.

~(O)

= 0,

(16)

478

FUNCTIONS OF COMPLETELY REGULAR GROWTH

J

00 e-e/e(r/2)

[ APPENDIX VIII

d,,[ qo, r)] < 00 rP(r)

(17)

,

converges Jor every e > 0, then u(z) belongs to class c). 4. IJ (16) is satisfied and

J

oo

-£/.(r/2)

re

d,,[ qo, r)] rp(r)

<

(18)

00,

then u(z) belongs to class d).

In particular, (17) is satisfied if e(r) = o(1lln In r), and condition (18) is satisfied if e(r) = 0(1 lin r). Statement 1 is a generalization to subharmonic functions of Theorem 6 of Chapter II, and also contains the corresponding result of Krasickov [1; 3] for entire functions. Grisin extended these results to subharmonic functions in the half plane 1m z > 0, with the stated formal order. He also investigated subharmonic functions of order zero in the half plane 1m z > 0. This case is essentially different from the case p > 0. 4. The second term of an asymptotic formula

Let the zeros of the canonical product J(z) be positive, let n(t) be the counting function of the zeros, and let net) = !ltP + !l)tP1 + fP(t), (19) where p > p) > [p], !l > 0, and fP(/) = O(/PI) as 1 -+ + 00. Then it is easy to see that for arbitrary 8 > we have the asymptotic equation w!lrP -p(fJ ) w!l)rP1 e'PI - (fJ - f r) + 0(rP1 ), lnJ(re ill ) = -.--e' - f r +. (20) sm wp sm wp)

°

as r -+ 00, uniformly for (J E [8, 2w - 8]. It is natural to ask whether (20) holds uniformly in the plane except on a "small" set. Logvinenko [2] gave a positive answer to this question. Moreover, if we assume even somewhat less about fP( I), namely that

r

2T

T-+

)T IfP(tW dt = O(TPlq+),

for some q

00,

(21)

> 1, we have the asymptotic equation InJ(reill) =

~!lrP

sm wp

eip(fJ-tr)

+ ~!l)rPl

sm wp)

eiP1(fJ-tr)

+ t/I(reifJ),

(22)

where t/I(re ill ) = O(rPl) as r -+ 00, uniformly for (J E [0, 2'IT], provided that z = re ifJ is outside an exceptional set E = Ej" This exceptional set is the union of a countable collection of rectangles x~ < Re z < x;:, 11m zl '" y" (n = 1, 2, ... ) that satisfy

Sec. 4]

479

SECOND ASYMPTOTIC TERM

a) x~ > 0, n = 1, 2, ... , b) Ix,;
>

1, we have the following asymptotic estimate

T_oo.

(23)

Finally, there is a converse theorem. If an entire function f(z) satisfies the asymptotic equation (22) in the complex plane except on the small set E" and the zeros of this function lie on the positive real axis, then the counting function n(t) has the form (19) with qJ(t) = 0(tP1). In the theorem of Valiron and Titchmarsh (see Appendix Vll, assertion 2) if all the zeros of an entire function of order P lie on the positive real axis and

lnJ(-r)

'lTtl. sm 'lTP

= - . - rP

+ o(rP ),

(24)

then n(t) = tl.t P + o(t P). However, a two-term asymptotic formula 'lTtl. InJ(-r) = -.-rP sm 'lTP

'lTtl.1

+ -.--rPI + O(rPl) sm 'lTPI

(25)

for J(z) with only positive zeros does not imply the two-term formula (19). A number of papers have been devoted to the behavior of n(t) when (25) or similar formulas are assumed. Tjan [1] and Anderson [1] showed that n(t) satisfies the sharp asymptotic formula net)

= tl.t P + o(tP/ln t),

t_oo.

Logvinenko [3] showed that even under the more restrictive condition that (20) is satisfied uniformly in the angle < 8 <; (J <; 2'IT - 8, the asymptotic equation (19) may fail.4 The situation is different if we require that (22) is satisfied on both rays (J - 0, 'IT. Logvinenko [3] proved the following theorem: Let J(z) be an entire junction oj nonintegral order with only positive zeros, let (22) hold Jor (J = 0, 'IT, where P > PI > [pI, and let Y,I(X) satisfy

°

f

1y,(x)j9 dx = o(TPlq+I),

T_oo,

T
on the real axis.

4For this circle of ideas. see also Subbankulov and Tolbaev [1], Tolbaev [1; 1]. Jurov [1], and Qapter VII of the book [1] by Subbankulov.

FUNCTIONS OF COMPLETELY REGULAR GROWIH

480

[ APPENDIX VIn

Then n(/) has Ihe form (19), and Ihe remainder lerm cp(/) - 0(1"·) as 1-+ 00 excepl on a sel of reialive measure zero. If also q > 1, lhen CP(/) salisfies (21). Logvinenko also proved theorems for two-term asymptotic formulas that are similar to the theorems of Chapter II of this book. One of these is as follows. THEOREM.

junclion n(/,

Lei f(z) be an enlire junclion of order P and lei lhe dislribution (JJ of lhe zeros of f(z) salisfy

(JI'

n( I,

(JI'

+ d l ( (JJ' (J2)/"· + cp( I, (JI' (J2)'

(JJ = d( (JI' (J2)/"

where 1 > P > PI > 2p/3 and cp(t, (JJ' (JJ = o(tJ,,·-lp), 1-+ 00, uniformly in and (J2' 0 < (JI < (J2 < 2. Then lhefollowing asymplOlic equation holds: InlJ(reill)1 = H«(J)r"

+

HI«(J)r"·

(JI

+ "'(reill).

Here H«(J) =

~

H I( (J) =

. '11' sm '1I'PI

sm '1I'P

r(J }(J-2fr

cos p«(J - '" - '11') dd(",),

r(J }(J-2fr

cos PI( (J - '" - '11') dd J(",),

d l (",) = dJ(O, "') for '" E [0; 2'11'J and d l (", + 2'11') = We have I"'(reiII)1 = o(r"·) if r does not belong to a cerlain exceplional set E = E, of relative measure zero.'

where d(",)

d l (",)

= d(O, "');

+ d l (2'11').

5. Derivadves and primidves of entire functions of completely regular growth

Gol'dberg and Ostrovskii [lJ investigated the derivatives and primitives of entire functions of completely regular growth. They proved that the primilive of an entire junction of complelely regular growth is again an enlire junclion of completely regular growlh,6 and described the set of values cp for which the ray arg z = cp is not a ray of completely regular growth for the derivative. Azarin [8J showed that this description is exhaustive. In order to present it, we introduce some notation.

SEntire functions whose zeros are on a single ray have been investigated by many authors. For a summary and references see Boas [4]. We note the following result of Ostrowkii [4]: Let j(z) be Oft e1lliTe jrmction who&e zeros are Oft the ray arg z - a. If lim ,.., In Mfi,) - ao, ,~oo

lim ,..,-1/2 In Mfi,) - 0, ,~oo

where P is Oft ;1IIege" then j(z) CtIIUIOI be bouNJed Oft thlll ray. Condition (.) is satisfied, in particular, for p < p

+ 1/2.

When p - 0 this implies

'This result was obtained independently by Gruman [1), but part of his proof is incoacIUIive.

Sec.

5]

481

DERIVATIVES AND PRIMATIVES

Let h(rp) be the indicator of a function J(z) that is of completely regular growth with respect to a proximate order p(r). Let S be the interior of the set of values rp for which h(rp) = 0 (when p <: j this set is empty). We remove from S those of its component intervals (a, fJ) for which at least one of the conditions l)fJ - a >'IT/p,

2) h~(a) = 0,

3) h~( fJ) .. 0,

is satisfied, and let S 1 be the set of remaining intervals. Let l(f') be the set of values rp for which arg z = rp is not a ray of completely regular growth for 1'(z); we recall that this set is open, by Theorem I of Chapter III, § 1. This open set can be described as follows. We have l(f') C SI' and if l(f') intersects a component interval (a, fJ) of SI for which fJ - a = 'IT/p then (a, fJ) C l(f'). In particular, it follows that the derivative of an entire function of order p <: j and of completely regular growth is also an entire function of completely regular growth. When p > j it can fail to be of completely regular growth only on the rays for which h(rp) = O. Gol'dberg and Korenkov [1; 2] investigated the asymptotic behavior of the logarithmic derivative of an entire function of completely regular growth. Their sharp asymptotic formula for the logarithmic derivative holds outside a collection of disks C(1j, Zj) of p.-density zero with I < I' < 2; that is, a set of disks such that

They established the following results. 1. Let J(z) be an entire function oj nonintegral order oj cOl'lJlletely regular growth with respect to a proximate order p(r) __ p, let a(1/I) be the angular density function oj its zeros, and let F(z) = J'(z)/j(z). Then Jor every 1', 1 < p. < 2, there is a collection K oj disks, oj p.-density zero, such that, when E C\ K,

n.

F(re.) = -'lTp/sin 'lTP ,-"<,)-1 X

fo 2"exp {-i1/l + i(p + 0(,-"<,)-1)

forO <: rp

< 2'IT.

1)(1 rp -

1/11 -

'IT )sgn( rp

-

1/1)} da( 1/1)

482

FUNCTIONS OF COMPLETELY REGULAR GROWTH

[ APPENDIX VID

2. If the order p = [p] then F(re itf ) = prP(T)-lexp(i(p - 1)'1') X {

8

+

i

102" [ 'I' -

l/I - w sgn( 'I'

-

l/I) ]exp( -ipl/l) d~1/I} + O(rMT)-I)

outside the exceptional set. For example, for the Weierstrass r-function this yields

r{z)

=

r{ 2ij1e itf

-

2iw 7\s~:,'Pa. } + o(r)

outside a set of disks of ,,-density zero. Here a. is the angle in the period parallelogram, and "11 = nt)· We note that a set of ,,-density zero can be rather large. For example, the domain bounded by the parabola x > 0, Iyl < x, has ,,-density zero for every " > I. Hence it is important to have examples that show that when" = 1 the asymptotic formula does not necessarily hold. Such an example is, as is easily seen, the functionf(z) = cosYz . For integral order the existence of an angular density is not a sufficient condition for completely regular growth. However, even in this case there are asymptotic formulas, but for orders different from those for which the angular density exists. For this kind of order see 16. 6. Some complements to the theory of entire functions of integral order

If all the zeros of an entire function of nonintegral order p are on the

positive real axis and

n(r) = Arp(T),

(26)

then, by Theorem 25 of Chapter I, when p is nonintegral we have the asymptotic equation

Inlf(rei')1

R::

~eip('-")rP(T) sm wp

(0

< IJ < 2w);

that is, the functions n(r) and In Mt
oc

a) ~ a;;" =

00,

fJ) ~ a;;" < I

I

'See GoI'dberg and Ostrovskii' (1). Chapter II.

IS.

00.

Sec. 6]

483

FUNCTIONS OF INTEGRAL ORDER

In case a) we have lnj(rei/J) = InJ,(rei/J) ,

r Pe ip8

+ -P

~

a.<,

a;t,

and by Lemma 9 of Chapter I lnj(rei/J)

~

_!!.[! - i(fJ -

'IT)]e ip8 . rP(')

P

+ rPe ip8 P

r' dn(/) .

Jo

IP

The last term, which we denote by g(r, fJ), can be transformed by integration by parts, and we obtain e ip8 g(r, fJ) ~ n(r)P

+ e ip8rP

f' I

_

II' +

dl I

(). P t

It is easily verified (by L'Hospital's rule) that

L(r) =

~' IP+~~P(')

is a function of slow growth and therefore rP

~'IP(I)-P-I dl =

r P,('),

where PI(r) is a new proximate order. It is easily seen that Consequently g(r, fJ) ~ lle ip8r P,(') and in addition lnj(rei/J) ~ lle ip8rP ,(').

rP,(')-P(I)

-+ 00.

(27)

In case /3) it is convenient to write the canonical product in the form 00

j(z) = e-SZP

II

G(z/ an; p),

where S = p-I~~a,7. This can also be written in the form

j(z) = J,.(z)exp [-!.!:...

L

a,7],

II

G(z/an; pl·

P a.>,

where

J,.(z) =

II all<;;r

G(z/an; P - 1)

a,,>r

By Lemma 9 of Chapter I we have (28)

The logarithm of the second factor,

g (r fJ) = _e ip8 r" .1'

foo dn(t)

p,IP'

484

FUNCTIONS OF COMPLETELY REGULAR GROWTH

[ APPENDIX VIII

can be transformed as in the previous case, and we obtain

9 (r, 9) = 1

_ei,,8 n(r) _ rpfoo dt . P r tP+I-p(I)

It is easily verified that in this case 91(r, 9) ~ _lle i,,8rP 1(r) and rP1(r)-p(r) ~ Thus we have the following proposition: If f(z) is a canonical product with integral exponent of convergence, zeros are positive, and if it satisfies (27), then lnf(re il ) ~ ±lle i ,,8rP 1(r),

00.

if its (29)

where

L (r) = rP1(r)-p =

{f.

r

I

dt t P+I-p(I)

foo

1

r

f.oo dt = tP+1-p(I) f.oo dt < when

when

dt tP+I-p(I)

00,

I

1

tP+I-P(I)

(30) 00.

In (29) the sign is + if the integral JOOt-p-1 +P(I) dt diverges and - if it converges. As in Chapter II we can deduce from (29) that when p is integral and an angular density 1l(9) exists, the canonical product satisfies the asymptotic equation lnf(re il )

~

±rP1(r)

r'

eip(fI-rl-) dll(I/I)

J'-2'1r

or lnf(re il ) ~ Ct.ei,,8rP1(r),

(31)

where Ct. = ± J~"'e-i"'" dll(I/I). Of course, this formula is of interest only when C11 is different from zero. As was shown in Chapter II, the condition CIl = 0 is necessary for an entire function of integral order with angular density 1l(I/I) to be of completely regular growth for the proximate order p(r). Lunc [1] has shown that if the angular density /l(I/I) exists and CIl = 0, one can augment the given set {ak} of zeros by a set {bk } of density zero in such a way that the new canonical product satisfies condition (B') of Chapter II, § I; that is, it is of completely regular growth with respect to the proximate order p(r).·

7. Upper and lower iodkators and their appIicatioD to the distnbution of the zeros with respect to their .......nts In order to study the connection between the growth of an entire function in various directions and the distribution of its zeros with respect to their 'The cue of func:ticms of zero order is exc:eptioDaL Griiin [ot) has investiptecl the theory of functions of order zero that are of completely regular powth in an aqIe.

Sec. 7]

UPPER AND LOWER INDICATORS

485

arguments as well as their moduli, it is of course not essential to assume the existence of an angular density for the set of zeros. We can, for example, describe the distribution of the zeros in terms of the upper and lower angular densities (32)

One can describe the growth of an entire function by means of the lower indicator as well as by the indicator. This notion was introduced in a special case by Leont'ev [9] and in the general case by Gol'dberg [4]. The lower indicator b,,(9) is defined as follows: remove an arbitrary CO-set from the plane, form

M9,

CO) =

lim

,,-e CO '-+00

and then take the supremum over all CO-sets, i.e.

M9)

= sup

M9,

CO).

(33)

CO

It is easily seen that if J(z) has no zeros inside an angle a < 9 < p, then for these values of 9 the lower indicator b,,(9) is simply equal to the corresponding lower limit (without preliminary exclusion of a CO-set). The ordinary indicator is sometimes called the upper indicator. Clearly the equation b,,(9) = h~9) means that f(z) is of completely regular growth. A number of papers have investigated the problem of estimating the upper and lower indicators of an entire function of finite order in terms of the upper and lower angular densities of its zeros. If p(r) denotes the proximate order in terms of which we have defined the upper and lower densities of the set of zeros, it is natural, in the light of the theorems of §6 of this appendix, to define the indicators in terms of the same proximate order p(r) when the order is nonintegral, but in terms of the proximate order pJ(r) defined by (30) when the order is an integer. To begin with, we can investigate entire functions whose zeros are aU positive, and estimate the upper and lower indicators in this case in terms of the upper and lower densities &and 4 of the set of zeros on the positive real axis. Theorem 12 of Chapter V can be considered as a result of this kind. We have only to replace z2 by z in the representation of the entire function to obtain an entire function F(z) of order P = ~ with positive zeros, and the set of zeros has &= I and 4 = o. Theorem 12 of Chapter V provides a sharp estimate for the indicator of such a function. In fact, an extremal function is constructed for which the

486

FUNCTIONS OF COMPLETELY REGULAR GROWTH

[ APPENDIX VIII

estimated value is obtained for each fJ E [0, 2'11'). Andrasko [I) obtained a similar result for entire functions with positive zeros and arbitrary order p, 0< p < l. Then Gol'dberg [I) obtained a similar result for functions of arbitrary positive order. Govorov (3) found, for < p < I, sharp estimates for the upper and lower indicators of an entire function with zeros on the positive real axis and given upper and lower densities X and ~. Finally Gol'dberg (4) obtained such estimates for arbitrary order.9 In the general case the estimates are rather complicated. The simplest are for integral p:

°

X +11

X-11

-rcos pfJ - ---y=lcos pfJl ..;;; ~(fJ) ..;;; h(fJ)

X +11

..;;; ---y=cos pfJ

a-a

+ ---y=lcos pfJl,

(34)

where equality can occur at any of the ..;;; signs. Naturally in attempting to obtain similar results in the general case, one does not require that the zeros of the entire function are all on one ray. The results have been worked out by Gol'dberg in a series of papers [2-5]. Let the set X C [0, 2'11') and let n(r, x) be the number of zeros of J(z) in the set {Izl = r; arg z E X}. Put

By analogy with the results of Chapter II we might suppose that we could make the transition from the case where the zeros are on a single ray to the general case by using an interpolation process on the upper and lower densities of the set of zeros. However, as is easily seen, the functions X(X) and ~(X) are, in general, not additive. We may, for example, take a set of zeros such that X(X) == I and ~(X) = for all intervals X C [0, 2'11'). Thus the problem of estimating indicators of entire functions leads to an integral of a new kind, corresponding to two nonadditive measures. Such an integral was introduced and investigated in detail by Gol'dberg [2, 4). Let A be a collection of Borel sets on the half-open interval [0, 2'11'), and let JA..(a) and p?'(a) be two nonnegative functions (measures) defined for all subsets a of A, with p?'(a) ;> JA..(a). We may suppose without loss of generality that JA..(X) and p?'(X) are defined for all subsets X of [0,2'11') by

°

j = 1,2,

fora EA.

'Other contributions to the same circle of ideas have been made by Leont'ev [9], KrasiCkov [2], Ostrovskii [41. Kondrayuk [II. [2). Baillette [1). Kondrayuk and Fridman [1. 2], and Azarin ['I.

Sec. 7]

487

UPPER AND LOWER INDICATORS

Let N denote the class of finitely additive measures p(X) that satisfy I'I(X) '" p(X) '" IL2(X) for all subsets X of [0, 2'17). It is easily seen that this class is not empty, provided that I'I(a) and lL2(a) satisfy the subadditivity conditions

+ I'I(a2)'

I'I(a l U a2) ;) I'I(a l )

I'ia l U a2) = 1'2(a l )

+ lL2(a 2);

the lower and upper densities of the set of zeros certainly satisfy these conditions. We define the integral of f(t) with respect to the measures I'I(X) and IL2(X) by

r2"f(t)d[ I'I(t); lL2(t)] = max [ r 2"f(t)dP(t)],

)0

p(x)EN

)0

(35)

where the integral on the right is taken in the Radon sense, and the existence of the maximum can be established. We also define the "lower integral"JO

r2"f(t)d[ I'I(t); lL2(t)] = min [ r2wf(t)dp(t)

)0

p(x)EN

=

)0

r

\

[2" [-f( t)] d[ I'I( t); 1L2( \ t) J.

-)0

Gol'dberg's estimates were established by the use of this integral. We quote some of them. Let I'I(X) and IL2(X) be measures on [0, 2'17). Let F( 1'1' 1L2, p(r» be the class of entire functions of order p(r) when p is nonintegral, and of order PI(r) when P is an integer, for which (36)

If P is an integer, we have

- fo2w [ -cos p( 0 - 1/;)] d[ 1'1 (1/;); 1'2(1/;) ] <;

h(O) <; fo2wcos p(O - I/;)d[ 1'1(1/;); 1L2(I/;)]

(37)

for all f E F( 1'1,1'2' p(r».

lOWe note that the measure th(t) for which the maximum of the Radon integral on the right of (35) is attained depends on the integrand f(t). In his papers, Gol'dberg gave a more general definition of the integral with respect to a nonadditive measure. He also gave a different but constructive definition, rather complicated to state but adequate for his investigation. The equivalence of the two definitions was proved by Levin, Macaev and Ostrovskll when I'I(X) - O. This proof is reproduced by Gol'dberg ([1), Theorem 2.10). He also lot) proved the equivalence for I'I(X) "'" O.

488

[ APPENDIX

FUNCflONS OF COMPLETELY REGULAR. GROwm

vm

These inequalities are sharp in the sense that the class F( 1'1' 1'2, p(r» contains functions for which equality holds either on the left or on the right. Moreover, there is anf E F( 1'1' 1'2, p(r» such that the right-hand inequality is satisfied simultaneously for all arguments. If P is noninlegral, we have Ihe inequality

h(lJ) ;> -'II

(2,,[

)0

cos p(I~ SlO

"'I -

'liP

'II) ]d[ I' ("'); 1'2("')]

(0

III;

I

IJ

< 2'11).

When I'I(X) == 0 the indicator of every function in the class F(O, 1'2, p(r» satisfies the inequality (2"

(aJ

h(lJ) 111;)0 Ip-p-Idl)o K(/, IJ - ",)d{O; 1'2("')} where K(/, IJ) (0

III; I III;

00; 0 III; IJ

(p

= [p]),

(38)

< 2'11) is the kernel equal to

K(/, IJ) = cos pO - I cos(p + 1)1J l-/coslJ+/2 at the points (I, IJ) at which both inequalities

IG(e il fl,p)1 > 1

! IG(e fl,p)1 < ° il

and

are satisfied, and equal to zero at the remaining points. On the other hand, one can construct an entire functionf(z) E F(O, 1'2, p(r» such that (2"

(aJ

hfO) ;»0 d{O, 1'2("')})0 tp-P-IK(t,O - "') tit

(39)

for all o. The integrals on the right-hand sides of (38) and (39) differ only in the order of integration, but they are not necessarily equal, and hence (38) is not always sharp, as can be shown by examples. However, if I'2(X) is an additive interval function, we can replace d{O, 1'2("')} in both integrals by dl'2("') and take the corresponding integrals as Lebesgue-Stieltjes integrals. The repeated integrals are then equal, and (38) becomes sharp. In particular, this will be true if I'2(X) is taken to be the "maximum density" of the set of zeros. Let us consider this case in more detail. P6lya [1] introduced the concept of maximum density for sets on the positive real axis: ~ = lim lim k.j.1 , ..... aJ

n(kr) - n(r) . (kr)p(kr) - ,,<,)

This concept can be generalized as follows to sets more generally distributed in the complex plane. We define the maximum angular density of a plane set to be X(",) =

lim maxl'h+I-oI-A.....o

{ ~I lim lim n(kr, th, th+l~ - n(r, th, th+l) }, j-O k.j.1 ' ..... 00

(kr)p( )- ~,)

Sec.S]

MISCELLANEOUS RESULTS

489

where 0 = 1/10 < 1/11 < ... < 1/1" ... 1/1. Evidently ;1(1/1) ;> ~I/I) and ;1(1/1) is an additive interval function. II We mention two other cases in which sharp upper estimates can be obtained for the indicator: I) I'I(X) = 0 and I'2(X) = ll. = const; that is, for the class F( 1'1' 1'2, p(r» given a global (not angular) density for the set of zeros; 2) I'I(X) and I'2(X) are additive interval functions and satisfy some additional conditions. In the special case I'I(X) = I'2(X) we obtain (2.04) of Chapter II. We also note that inequalities of the same kind as Gol'dberg's for entire functions were obtained by Fainberg [1,2] for functions holomorphic in a half-plane; instead of ordinary angular density he used the argumentboundary density (see 12 of this appendix).

8. MisceUaneous results from the theory of functions of completely regular growth Azarin [10] established a connection between regularity of the growth of an entire function and regularity of growth of its coefficients: Ck(r) =

2~

i:

lnl J(rei1/l)l e -i1a/- dI{I,

k = 0, ± I, ±2, ... , by expanding Inlf(re~)1 in a Fourier series. His result is as follows: A necessary and sufficient conditionJor J(z) to be of completely regular growth with respect to the proximate order p(r) is that the limits

.

11m

r-+oo

Cir)

~) = r-,r

Ck ;

k = 0, ± I, ±2, ... ,

(40)

exist. In this case the indicator of f(z) is represented by the series

It is important that the conditions (40) are independent. Kondraljuk [3] extended this result to the more general orders introduced and investigated by Rubel and Taylor [1]. Let ~(r)too and let ~(2r) < M~(r) for all r> O. An entire function f(z) is said to be of completely regular growth with respect to ~(r) if IIWe have the following fact. Among aJ'-Xt8 containing the gWen xt and haoinB IlIf(pI/tu deluitia, there is a set '!JR. SllCil that ~(+) == 6<+). Sharp estimates for the indicators of entire functions with given maximum and minimum densities were obtained by Kondratjuk [I].

490

FUNCTIONS OF COMPLETELY REGULAR GROWTH

[ APPENDIX VIII

,li~ ~(lr) {~1l1nlj(re~)1 em } exists for all intervals l2 0 <; a < fJ <; 277. As indicated in the footnote to Theorem 2 in Chapter III, 12, for ~(r) = rl'<.') this definition is equivalent to the one given in the text. In this more general case the theorem can be stated just like Azarin's theorem. The theory of functions of completely regular growth can be generalized by considering the growth of the function on 10garithInic spirals instead of on rays. Correspondingly, the angular density of the set of zeros can be replaced by the density of the set of zeros between two spirals. Balasov [1, 2] considered more general curves

v.,(a, b)

{t; t

=

= Itlei[Y(III)+'P), a <; It I <;

b,

-00

< 'P < oo}

for which the liInit

lim

x-++oo

xy'(x) = c

exists. He calls these curves of regular rotation. When y(x) = c In x they are logarithmic spirals. Introduce the notation

D,Y(a, fJ) = {z; z E

~(a,

D~(a,

fJ)

x), a <;" <; fJ,lzl <; r},

= DY(a, fJ),

and the function

MY(r, a, fJ) = max {j(rei(Y(,)+.»}. a<{}<11

J

The order and type are defined in the natural way, and the indicator is now l3

h/(") = lim {r-P(')lnlj(z)l}. zEEl

The angular density of the set of zeros of an entire function is defined by

AY( a, pQ) = 1·1m nY(r;I'<.a,) fJ) , '-+00 r' where n Y(r; a, fJ) is the number of zeros of j(z) in D,Y(a, fJ) and Loa

~Y = f

lim rP·P(')eiPY(')[c '-+00

+.!. ~

P 1,,*1 <,

ai!'].

12We also note the papers of Cemoljas [1, 1), who considers orders more general than proximate orders. He assumes that Iim,._oo p(r) - p < 00, Iim,_oa p(r) - >. > 0 and rp'(r)1n r_ O. He carries over the theory of functions of completely regular growth to such "oscillatory" orders, which are somewhat more general than Boutroux orders. 13-Jbis indicator was studied by Macintyre [1) in the case when

fI

is a logarithmic spiral.

Sec.S]

MISCELLANEOUS RESULTS

491

The set of zeros is called y-regularly distributed for a nonintegral p if the angular density IlY exists for all a and p except perhaps for a countable set; and for integral p, if the limit 8/ exists. A function f(z) is said to be of y-completely regular growth if, after exclusion of a CO-set, the limit hl(~) =

lim

{r-P(r)lnlJ(z)l}

zefi.zE CO

exists. With these definitions, we have the theorem: For an entire junction with proximate order p(r) --+ p to be of y-completely regular growth, it is necessary and sufficient that the set of its zeros is y-regular/y distributed. There are also asymptotic formulas of the same kind as (2.03), (2.04) and (2.06).

LIST OF IMPORTANT IDEAS AND THEOREMS Proximate order. A function p(r) (r (I) lim r -+ co p(r)

> 0) is called a proximate order if

= p exists,

(2) lim r-+ co rp'(r) In r = O.

If -

lim

InM,(r)

,-+00

(

rP r)

+0, co

for an entire functionf(z), then p(r) is called the proximate order off(z). Every entire function of finite order has a proximate order. A slowly increasing fllllCtion L(r) is a positive function for which lim L (kr) = 1 L (r)

r-+oo

uniformly on any segment 0 < a <; k <; b < co. If p(r) satisfies conditions (1) and (2), then r P (r)

= rP L (r)

(p. 32).

A function designated by L *(r) is one that can be expressed in the form In L· (r) = 32 (In r) - 31 (In r),

where the {}i(r) are concave functions satisfying the conditions lim 3, (x) = co, ~-++oo

lim

3, (x)

~-++oo

X

= 0,

· 11m aI-++ 00

3"i (x)

-,-_=.0 &i (x)

Every function L *(r) is slowly increasing. A function p*(r), defined by the relation rpe(r) =rPL·(r),

is called a strong proximate order (p. 41). Every entire function of finite order l.'5 a strong proximate order (pp. 35-41). Deality. Let 91 be any point set in the plane, let n(r) be the number of points of 91 in the circle Izl < rand n(r, {}, 6) be the number of points of 91 in tht sector Izl < r, {} < arg z <: 6, and let p(r) be a proximate order. 492

LIST OF IMPORTANT IDEAS AND THEOREMS

493

The density and the upper density (a similar definition yields the lower density) of the set 91 relative to the exponent p(r) are defined to be fl = fl =

11m n (r)

(density).

11m n (r)

(upper density).

r~oo rP (I')

r~oo rP (I')

The set 91 has an angular density, if fl (&. 6) =

11m n (r. &. 0)

r~oo

,p(r)

exists for all {} and () with the possible exception of an enumerable set. In this case the relation fl (0) - fl (&) = fl (&. 0) defines a nondecreasing function ~«() with an arbitrary additive constant (pp. 89-90). The linear density p*(C) of a set C of circles C; in the complex plane is defined to be p* (C) = 11m -1 ~r rio r~oo

,

where r l is the radius of C; and :E denotes the sum over all the circles with centers in the circle Izl < r. If p*( C) does not exist, we introduce the upper linear density r

-

p*

= ro+co 11m -1 ,

~r

'J

of the set and the lower linear density p*(C). A set of circles with zerp linear density is called a CO-set (p. 90). The relative measure of a set E of positive numbers is the limit m* (E)

=

11m mes (E') , ,

r~oo

where Er is the intersection of E with the interval (0, r). The upper and lower relative measures are defined similarly:

m* (E) =

11m mes (E") • r~oo'

m* (E) =- lim mes (E") • -

r:+ao'

The symbol EO is used for a set of zero relative measure (p. 96). Sets of points 91. regularly distributed for the exponent p(r). I. p not an integer. A set 91 is said to be regularly distributed for the exponent p(r) if it has an angular density for this exponent (p. 90). II. p an integer.

494

LIST OF IMPORTANT IDEAS AND THEOREMS

A set 91 is said to be regularly distributed for the exponent p(r) if it has an angular density for the exponent p(r), and if the limit

a=

11m - 1. { c L (r)

r-+co

+ -P1

~

-a'I}

(a" E 91)

la,,1 <:r "

exists (is finite) for any value of the constant c (p. 92). The zeros of an entire function are regularly distributed if the set of all these zeros is regularly distributed for the exponent p(r), where p(r) is the proximate order of this function. R-sets. A regularly distributed set {a .. } is called an R-set if (C) there is a number d > 0 such that the circles of radius

r,,=dlanl

l-.!.pUa I) 2 "

with centers at a.. are non intersecting, or if (C) all the points a .. lie in a finite number of nonoverlapping sectors q>; < arg z < q>j (j = 1,2, ... , m), and la"+Ljl-lan,jl>dlan, jl

l-p(la

ni

I'

(

p.95

)

for any d > 0, where all the points a..1 in each of the sectors are numbered in order of increasing absolute value. The indicator of an entire function. The indicator of an entire function fez) of exponential type is h (6) ,. hf (0)

=

11m In II (re") 1

r

r-+co

The indicator of an entire function f(z) offinite order p and of normal type is h(O)=h (0)= 11m f

l'

Inl/(re")1

-+co

r'

(p.51).

The generalized indicator of an entire function f(z) of proximate order p(r) is In II (re"') I h (0) = ,~..(b) = 11m ".1') •

1'-+ co

(p.70).

<

The indicators of functions that are holomorphic in some angle at <; arg z P are defined similarly. The characteristic property of an indicator (and of a generalized indicator) is its trigonometric convexity. A trigonometricaUy COIlTex function is a function satisfying the inequality h

(Ot> sin p (~ -

Os)

+ k (OV sin p (Os -

at> + h (Os) sin p (°1 - OV < 0

< ..!:. "I < lit < 6. p We will use the notation s(8) = h'(8) + pi r h(q» for

0. -

01

(p. 5~~. dq>, $(8, IJ)

= s(8) -

s(IJ).

LIST OF IMPORTANT IDEAS AND 11IEOREMS

495

For a function h(9) to be trigonometrically convex in some interval, it is necessary and sufficient that s(9) be nondecreasing in this interval. When p is not an integer we have I

1& (8) =

1

2p sIn 1I:p

f

cos P (8 -

~ ~ 11:) ds (4)·,

1-2"

when p is an integer

h(8)=~p

8

f

(0-+) slnp (8-t)ds('l-)+Acosp8+Aslnp8 (P·60)·

8-2&

An indicator of the form a cos p8

is caned sinusoidal.

+b sIn p8

THEOREM. For any strong proximate order p*(r) and any trigonometrically convex, periodic function h(9) with period 2"" there is an entire function with proximate order p*(r) and indicator h(9) (p. 94).

When p is not an integer, a similar assertion is valid for any arbitrary proximate order p(r). A strong proximate order can also be replaced by a proximate order for integral p if h(9) ;> 0 (0 <;; 9 < 2",). Supporting funetiODS. The supporting function of a convex region G is the function k (8) == sup (x cos 8 sIn 0), _Eli while the straight line I. with the equation

+,

x cos 8 + , sIn 8 - k (8)

=

°

is called the supporting line. The relations k (OJ sin (~ - fie) 01
+

k (8 211:) =k (fI), k (II,) sin (fI, - flJ k (fla) sin (81 - "') <: 0, 0.-0,.<11:, 02-01<11: (p.76)

+

+

are characteristic of a supporting function. Let the region G contain the origin, and let s be the arc length of the boundary of G measured counterclockwise from some fixed point to the supporting point. Then 0 is a nondecreasing function of s. At angular points of the boundary O(s) is discontinuous, while for straightline segments of the boundary O(s) is constant. The arc length along the boundary between the points of the support lines I... and I, is

8(~ +0) - 8 ( ... -0) -k~ (~) -k~ (11)+

,

f •

It (II) dI.

496

LIST OF IMPORTANT IDEAS AND THEOREMS

The radius of curvature ds

R= d8

exists for all values of 8 with the possible exception of a set of zero measure, and R

We also have

= It' (8) + " (8).

k(8)-

f•sin

(6-IjI)ds(Iji),

o

while, when the origin is displaced to any arbitrary point, k (8)

= A cos (6 -

80)

There is also another representation: "(6)

= B cos (6 -

60)

0/) ds (+).

o

,

J

+~

•

+ f Sin (6 -

(8 -;) sin (6 - ;) ds ('H (p. 81).

8-1r:

The iadicator diagram. The indicator for any entire function of exponential type is the supporting function of a bounded, convex region I which is called the indicator diagram of the function. The eonj.....te diagram of an entire function of exponential type co

fez)

=~ ~ an z'I III o

is the smallest region containing all the singularities of co

,(z)

= ~"::l. o

The function (}'(z) is called the Borel transform of/(z). THEOREM. The conjugate diagram of an entire function of exponential type is the mirror reflection of its indicator diagram in the real axis.

F.aioas of coaapletely regular powtIa. A function F(z), holomorphic in the sector ex < arg z < fl, is said to be of completely regular growth on some set of rays Ra (!JI a set of values of 8) if a set 4 of zero relative measure exists such that for r E 4 and r _ 00 the function (6)

h " r

= In I F(re·')1

r'

(f')

tends uniformly to h,.(8) for 8 E!JI. If a function F(z), holomorphic and of proximate order p(r) inside the angle (ex, {J), is of completely regular growth on some set of rays Ra in the angle

LIST OF IMPORTANT IDEAS AND THEOREMS

497

defined by (ex < 81 <; 8 <; 6. < (J), then it is of completely regular growth on the set of rays Rjjj. An entire function f(z) of proximate order p(r) is called an entire function of completely regular growth if there is a set EO of zero relative measure such that, when r - 00 with r E EO, the function h

(0)

P. r

= In I/(re")l rP(r)

tends uniformly to h,(6). The expression lim~_cof(r) will denote the limit of 00 without taking the values in some V-set (P.139). Let

fir), if it exists, when r -

f r

.!p. (0) _

In f F;te4') I dt

(p. 143).

° If F(z) is holomorphic and of proximate order p(r) in the interior of some angle (ex, (J), and if it is of completely regular growth on some ray arg z = 8 in this angle, then 11m r- P(r).!p(O) = p- 1h p (0). r ... oo

Conversely, if this equality holds on all the rays in some angle, then F(z) is of completely regular growth in this angle. THEOREM 1. Let p(r) be a proximate order of growth and let p = liMr_co p(r) be nonintegral. Let the set {arc} have the angular density d(8) for the exponent p(r). Then if we consider only values of z = reitJ which do not lie in a certain set C of circles of zero density (a CO-set), the entire function

with P(Z)=CO+C1Z+ ••. +cpZ~

(p

= [~J).

lGIisJies, uniformly in 8, the relation m In I/(rel') .... (r) I

II

r ... oo

, where tt H,(6)=_l s n ttp

f•

r·

-= H,(0).

cosp(6-I!,,-tt)dA(t) .

(p.91).

I-Is THEOREM

2.

If P is an integer, the conclusion of Theorem 2 remains true with I

H,(6) =

f (.~ -6) I-Is

Sin p <~-6)dA <+) +~ cos p (6 -Ito)

498

LIST OF IMPORTANT IDEAS AND THEOREMS

if we also lJ

impose condition II in the definition of regularly distributed sets with c p (p. 91). Iff(z) has regularly distributed zeros, then

= Tei"O and with c =

It, (II) =- H, (II). For p = I, the angular density of the zeros of fez) in the angle (D,O) is (211')-1 times the arc length of the indicator diagram between the supporting points for the supporting lines I. and I", these lines being perpendicular to the rays corresponding to the two angles (p. 93). The converse theorems are also true. We have the following COROLLARY. For an entire function f(z) of proximate order per) to be of completely regular growth, it is necessary and sufficient that the zeros of this function be regularly distributed for the exponent per) (p. 158). CoROLLARY. An entire function of exponential growth can be expressed in the form

II<

fez) =z,neGHb 11m B-+""I at

~vpe

(1-~) B

and completely regular

(p.158)

lit

Entire functions of completely regular growth are those entire functions, with a given proximate order and a given indicator, which have the greatest possible lower density of zeros Ito

f

11m nl Vi) =..L hI (0) dO. 21t 0

- - rP r

r-+oo

For all other entire functions, the left hand side is smaller than the right hand side (p. 173). In §§ 2 and 3, Chapter IV, there are corresponding uniqueness theorems for functions holomorphic in some angle. 'I1Ie iadlcator o(tIle product o(two (1IDdioDI. Iff(z) and 9'(z) are holomorphic and have the same proximate order in the angle (11, (J) and if one of them is of completely regular growth, then the indicator of the product of these functions is equal to the sum of indicators of the separate functions (p. 159). CoROLLARY 1. If F(z) is holomorphic and of proximate order p(r) in the angle (11, fJ), and if the entire function cp(z) ;s such that

11m

,.-+00

In M, (r) ,.,

(,.,

= 0,

then the indicator of the product of these functioIU in the angle (Il, (J) is the same the indicator of F(z) (p. 160).

Q8

499

LIST OF IMPORTANT IDEAS AND THEOREMS

The caDOmeal fUIICtion of a regularly distributed set. Let 9( denote the set

{at}, regularly distributed for the proximate order p(r). For nonintegral values of p we write 00

fez, 91) =

II a(.!...; p) k~1

(p

= [p),

ale

and for integral p

fez, ~) =/A'~

co

II a(.!...; p). k=1

ale

Here c.,4 is the constant in the expression for 6. If this constant is nonnegative, we write c.,4 == O. The function Hl-fJ) (see the discussion of functions of completely regular growth) for the set 9( is called the indicator of 9(, and is denoted by H.(9) (p.92). The Lagrange interpolation formula. Let the interpolation points 9( = {a.} be regularly distributed for the proximate order p(r), letj(z, 9l) be the.canonical function, and let H.(9) be the indicator. For brevity, we setj(z) == j(z, 90. The series co

~

.t.J

,.=1

fez) bra

is called the Lagrange interpolation series. If 9( is an R-set, and if for any e > 0, N In I b,.1

(*)

(z - a,.) f' (a,.)

> 0, and n > N we have

< [H. (tjI,.) -1]1 a,.IP(J CI,.!)

(tjI" =- arg a,.).

then the Lagrange series (*) converges uniformly in any bounded region to an entire function with indicator smaller than H.(9) (p. 195). The function F(z) defined by the series (*) is not the only function satisfying the conditions of the interpolation problem and the inequality hP(9) H.(9). The general solution is

«

~ (z)

= F (z) +

18

(z) f (z).

where c.u(z) is an entire function of minimal type for the proximate order p(r). With the additional condition 11m

, . (z) I

I_I ~oo If(z) I

=0

outside the exceptional circles, we have c.u(z) == 0 and CIl(z) is uniquely defined (p.198). If the points a. form an R-set 9( with exponent p(r), then any entire function F(z) satisfying

h, (8)

< H. (8)

500

LIST OF IMPORTANT IDEAS AND THEOREMS

can be expressed in the form co

F( ) _~, z - ~

,,=1

F (a,,) I(z)

I' (a,,) (z -

a,.) ,

the series converging in any bounded region. If the requirement that {a,,} be an R-set is discarded, and it is only assumed that ~ is a regularly distributed set, then F(z)

==

11m ~ /(4,.)1(6) B~co ~ I (a,,)(z-a,,) B EE"ICI.I
(p.198).

If ~ is an R-set with exponent p(r), then an entire function solving the interpolation problem btl == F(a..} and satisfying the condition hF(8)~H. (8)

will exist if and only if 11m

In I btl I

,,~CO H. (1jI,,) I a" t( I II" I,

~ 1.

1be spectra of entire functiollS that can be approximated by exponential polynomials. For any bound.ed, convex region I, the expression II FIII

=

sup r>O. 0<8<2..

I F (reiD) I e-lIa(O,r

can be used as a norm for entire functions, where h(O) is the supporting function of the region I, symmetric relative to the real axis. The complete space E[ of entire functions is defined correspondingly (p. 288). If an entire function F(z) can be approximated in the norm of B[ by exponential polynomials

with exponents from the boundary of /, then the spectrum of the entire function F(z) in B[ is the name given to the set of points A on the boundary of I for which the quantity II (l) == 11m a" (A) ft~CO

is not zero. For other equivalent definitions of the spectrum, see Chapter VI, pp. 290 and 291. THEOREM.

If the

smallest convex region containing the whole spectrum of

E{z) coincides with its conjugate diagram, then E{z) is of completely regular

growth.

501

LIST OF IMPORTANT IDEAS AND THEOREMS

In particular, if the numbers A" are on the boundary of a convex, bounded region, then the entire function 1.

00

f(z)r= ~ line".

"=1 of exponential type is of completely regular growth. FunctioDS of the class [~]. A function of this class is almost-periodic with its whole spectrum in an interval of length 1~, the ends of the interval belonging to the spectrum. Such a function is an entire function of exponential type, the conjugate diagram of which coincides with the smallest interval containing its whole spectrum. For an almost-periodic, entire function of exponential type to be in the class [~J, it is necessary and sufficient that all its zeros be in a strip Iyl <; H parallel to the real axis (p. 270). The formula n (t)

= -11:A t +

II)

(t)

holds for the zeros of a function of the class [~1, where n(I' is the number of zeros of the function in the rectangle 0 < x <; I, Iyl <; H, a'ld c.o(l) is bounded (p.271). FunctiODS of the class A. A function of this class satisfies the condition

~llm ~J 0 (or 1m z < 0) are defined similarly (p.223). An entire function of exponential type belongs to the class A if the integral B

f o

In I F(x) F(-x) 1 l

+x

I

d .C

is bounded for all R > 0 (pp. 225 and 228). (a) For an entire function f(z) of expollential type to be of completely regular growth and to belong to the class ..4, :t is necessary and sufficient that (I) the integral

f

00

o

In If(x)f(- x) , d 1 Xl x

+

exists and (2) h,(O) + h,(1r) = O. (b) The conditions (I) and (2) are sufficient to ensure that a function F(z). bolomorphic and of exponential type in the half-plane 1m z 0 (1m z <; 0), belong to the class A and be of completely regular growth.

>

502

LIST OF IMPOR.TANT IDEAS AND THEOR.EMS

(c) Another sufficient condition for an entire function (or a function holomorphic in a half-plane) to belong to the class A and to be of completely regular growth is the existence of either of the integrals

f

CD

CD

In IF(x)1 d x 1 +x3

d

x.

-co

-co

THEOREM.

F(x)1 f In+11 +x3

or

If an entire function f(z) belongs to the class A, and if

where R(z) is a polynomial and co

~ 1a

11 Il IJl + ", (a,l)

1

<

00.

then (1) f(z) is of exponential type, co

(2)

f

In+ I/{x) 1 xli

+

Id

X

<

00.

-CD

It follows from (I) and (2) thatf(z) is of completely regular growth and that its indicator diagram is a segment of the imaginary axis. 11ae class HB. An entire function w(z) belongs to the class HB (HB) if (I) it has no zeros in the closed (open) lower half-plane, (2) for 1m z < 0 it satisfies the inequality 1111 (z) I :> I fI) (z)

I.

For an entire function cu(z) to be in the class HB, it is necessary and sufficient that it be in the class A and be expressible in the form

II (I - ...!..) e BP,,(2...) 4". Go

.. (z) _ z-e" (.)+1 ('HI)

"-1

II"

where" ;;. 0, u(z) is a real entire function, and

R ~ (II" + 16&:) z" .. ~ IIlZ". TIIBoREK. Let • (z)

where

== P (z) + IQ (z),

LIST OF IMPORTANT IDEAS AND THEOREMS

503

and

are real entire functions. For w(z) to belong to the class HB, it is necessary and sufficient that the following conditions be satisfied: (I) the zeros of p(z) and Q(z) are all real, simple and interlacing; (2) the entire functions u(z) and v(z) and the Weierstrass exponents Pn(zlaJ and Pn(zlb,,) satisfy the relation <XI

~ [p,,(:'J-Pn(~)]=O;

u(z)-fJ(z)+

n=-cx>

(3) the constants A and B have the same sign.

When the first two conditions are satisfied, the last condition is equivalent to the inequality Q' (x) p(x) - Q (x) P' (x) >0

(-oo<x
A real pair of entire functions is the name given to two real, entire functions p(z) and Q(z) if they have no common zeros and if any linear combination ",P(z) + pQ(z) with real", and p has no complex zeros. If p(z) and Q{z) are permitted to have common zeros, then they are called a generalized real pair. THEOREM. For an entire function w(z) = p(z) + iQ{z) to belong to the class HB (HB), it is necessary and sufficient that p(z) and Q(z) be a real pair (a generalized real pair) and that Q'(x.,)p(xo) - P'(x.,)Q(xo) > 0 for some real point xo.

1be class P. An entire function w(z) of exponential type is said to be a function of the class P if (I) it has no zeros in the open lower half-plane, (2)

h. . (;)-
(p.320).

The quantity d., = h.,( -Tr/2) - h.,(Tr/2) is called the defect of a function of the class P (p. 320). _ The class P consists of entire functions of exponential type of the class HB(p.320).

THEOREM. For an entire function of exponentilll type CJ)

(z)

10 belong

= P(z) + lQ (z)

to the class P, it is neceuary and sufficient that (I) the zeros of p(z) and Q(z) be real and interlacing, (2) the indicator diagrams of p(z) and Q(z) coincide, and (3) Q' (x) p (x) - Q (x) P' (x) 0 (pp. 321-322).

>

504

LIST OF IMPORTANT IDEAS AND THEOREMS

The class p.. This is the class of all entire functions in H B expressible in the form III

" (z) = e- Y

1111 (zj,

where the genus of w 1(z) is not larger than one and" ;> O. THEOREM. For an entire function (O(z) to be of the class p., it is necessary and sufficient that it be the uniform limit of a sequence of polynomials, where these polynomials have no zeros in the lower half-plane (p. 334).

An N-polynomial is a polynomial having no zeros in the lower half-plane (p.331). Majorants. A function w(z) of HB is said to be a majorant of j(z) if

1111 (z) I ~ If (z) I,

1(J)(z)I~If(z)1

for 1m z < O. Majorants of the class P or p. are called P-majorants or P*-majorants, respectively. The above definition of a P-majorant is equivalent to the following: A function w(z) of the class P is a P-majorant for the entire function fez) of exponential type if (I) the elCponential type of w(z) is not lower than that off(z), (2) /f(x)/ .;; /w(x)/ (- 00 < x < 00) (p. 362). Admissible classes. A subclass T of the class H B is called an admissible class if, when w(z) E T and fez) is subordinate to w(z), it follows that w(z) + fez) E T. The classes HB, P and p* are admissible classes (pp. 350 and 355). A ~rOperator is a homogeneous, additive operator, defined on the linear span of some T-c1ass of functions and mapping this class into itself. A ~ p-operator is denoted simply by ~, while a ~ p.-operator is denoted by ~*. A ~T-operator is said to be continuous if it maps every sequence of functions of a class T, uniformly convergent in every finite region, into a sequence of functions from T convergent in the same sense. THEOREM. If T is an admissible class, then a (}3T-operator preseTt'es subordination. More precisely: If w(z) E T andf(z) is subordinate to (majorized by) w(z), then ~T[f(Z)] is an entire function subordinate to the function ~[w(z)] (p.356). THEOREM.

The operation of differentiation is a ~- and a ~·-operator

(p.356). THEOREM.

Let a subclass Q of the class HB satisfy the conditions

(I) from w(z) E Q itfollows that eT~w(z) E Qfor any real T,

(2) from w(z) E Q it follows that w'(z) E Q (i.e., differentiation is a operator).

~Q

LIST OF IMPORTANT IDEAS AND THEOREMS

505

Then Q E P*. In other words, p* is the largest subclass of HB satisfying the conditions (I) and (2) (p. 353). THEOREM. If w(z) is a P*-majorant of the function f(z), then w(t)(z) is a P*-majorant of I't)(z) (k = 1,2, ...), and if w(z) is not a polynomial and IW(k)(ZO) I = l/,k)(zO>I for some point Zo (1m Zo <: 0); then f(z) = clw(z) + caw(z) (lei I + Ic.1 = 1) (p. 356). THEOREM. If w(z) E P, if the exponential type of w(z) is not lower than that of f(z), and if If(x) I <: Iw(x)1 (- co < x < co), then I/,t)(x)I <: IW(k)(x)1 (- co < x < co). If w(x) is not a polynomial, then if If't)(xO>I = IW(k)(xO>I for some point Xo of the real axis, it follows that f(x) = Clw(X) + caW(x) (lcil + Ical = 1). For w(x) = Meia~, we obtain a theorem due to S. N. Bern~teIn: If f(x) is an entire function of exponential type not exceeding a and If(x) I <: M (- co < x < co), then 1f'(x)1 <: Ma, with equality only for f(x) = cleia~ + cae-ia~.

The class P is the largest class of majorants in the following sense. If w(z) E P and w(z) E P, then there is an entire function fez), of the same proximate order and the same type as w(z), satisfying the inequality If(x) I <: I(.(l(x) I (- co < x < co), and 1f'(xo)1 ;> Iw'(xo)1 for some real point Xo (p. 366). A sequence of multipliers of the first kind is a sequence of numbers "0' "1' "2' ... , such that, if all the zeros of the polynomial

"ft, ...

71

p (z) = ~ afcZ" lc=O

are real, then so are all the zeros of the polynomial 71

~ YfcafcZ" (p. 341). lc=O

The numbers "" (k = 0,1,2, ... ) either all have the same sign or have alternating signs (p. 342).

"ft' ...

THEOREM. For the sequence of numbers "0' "1' ••• , to be a sequence of multipliers of the first kind, it is necessary and sufficient that the series IX>

~ ~i z"

(O! = 1)

71-0

converge to an entire function cp(z), expressible in the form IX>

, (~) = e'I-

II (I +7ofcZ), It-I co

,,<00

where a::>O and ..." >0 (k= 1,2, 3, ... )and~ ...

(P.346).

1=1

r-operaton. These are operators that transform any entire function co

I

(z) = ~ a.,z1

".0

506

UST OF IMPORTANT IDEAS AND THEOREMS

into an entire f.unction 00

r [fez)] = _-0 ~ T""AZ

fI,

where )'.. (k = 0, 1,2, ...) is a sequence of multipliers of the first kind all with the same sign (pp. 340-344). THEOREM. r-operators give the general form of permutable with the operator z(d/dz). A JeaseD opentor transforms any entire function

~-

and

~·-operators

00

f(z)

=

~ iIZ~.

_=0

into the polynomial

" 1 " (z) = IIZo + ~

_=1

(1 - !) (1- !) ... (1 - k n 1)

ilZtzlt •.

Obviously J..(z) ~ /(z) for n - 00. A Jenstn operator is a r-operator (p.343). TIle dus HBI (H&J consists of the entire functions w(z, u) satisfying the conditions: (I) w(z, u) is not zero for 1m z <; 0 and 1m u <; 0 (1m z < 0, 1m u < 0); (2) for 1m z < 0, 1m u < 0 we have

UI (z, II) I :> 'UI

(z. u) I, , UI (z, u) I :> ,. (z, u) I, lUI (z, u),:> lUI (i, U) I.

ne class P z consists of the entire functions of exponential type belonging to the class HBz (p. 366). TIle class P: consists of the functions of Hils of the form ID

(z, u) == .-T"'- Ttl"

Ull

(z, u),

where Wl(Z, II) is of genus one for each of the variables when the other is fixed in the lower half-plane and )'1 ;> 0, )'1 ;> o. All H'-polyaomiaI is a polynomial of two variables P(z, u) which is not zero for 1m z < 0, 1m u < o. Such polynomials belong to Hils. THEOREM. For an entire function w(z, u) to be in the c/rus PI, it is neceuary and sufficient that it be the uniform limit of a sequence of Hi-po/ynomillll.

Me"""" ror

r.ctioal or IlIOn . . . . _ ......ltIe.l A function w(z, u) of the class Hils is said to be a majorant of the entire functionf(z, u) if, for 1m z < 0 andlmu < 0, I_ (z, u) I:> \!(i, II) I, I_ (z, u) I:> If(z, II) I, I-(z, II) I:> If(z, ii)(. ,_(z, 1I)I:>lf(z, u) I, • For simplicity we limit our diIcusIioa to fimctions of two variables.

507

LIST OF IMPORTANT IDEAS AND THEOREMS

Admissible classes. A subclass Tz of HB. is called an admissible class if, when ro(z, u) E T. andf(z, u) is subordinate to w(z, u), it follows that w(z, u) + f(z, u) E T•. The classes HB., p., and PI are admissible classes. mT.-operaton are defined in the same way as for functions of one variable. If T. is an admissible class, then a mT.-operator preserves subordination. THEOREM. Every continuous m*-operator for functiOns of the single variable z is also a m*-operator for functions of the two variables z and u.

De general form of the operaton m ..... m*. An operator K[f] applicable to entire functions is called regular if it satisfies the condition xr/,,] - K[j] for n - 00 uniformly in every finite region, where/,,(z) is the function obtained by applying the Jensen operator to f(z) (p. 373). Upper c:ontinllOll:' operaton. An operator K[f] applicable to entire functions of exponential type is called upper continuous if for every A > 0 there is a number B > 0 such that, if then

K[ fo (z)] 1,-B 1-'_0 (p.376). THEoREM. A regular m*-operator can be expressed in the form sup I K [I" (z)) -

ti* [f(z)]

= f(- Du) [, (z, u»1

,,.OJ

00

here f(- Du )[, (u)]

=~

00

(-1)'-4",(11) (u)

for fez) = ~anzra,

11=0

fI-O

and qi{z, u) is of the class PI and of genus one in u for 1m z

< 0 (p.373).

THEOREM. An upper continuous m-operator can be written in the form ti [fez)] =/(-DII) [, (z. a)] 1"=0.

where qi{z. u) belongs to PI and is of exponential type in z for 1m u < 0 (p. 377). THEOREM. A regular m*-operator permutable with the operation ofdifferenti. atlon can be expressed in the form ti (f(z)]

•

= F(D) [/(z)] •

re F(z) is a function of the class p* and of genus one (p. 379).

BmUOGRAPHY

AOMON,S. [1] Functions of exponential type i" an angle and singularities of Taylor .Jeriea, Trans. Amer. Math. Soc. 70 (1951), 492-508. AORANOVJC, P. Z. [1] 0" junctions of several fJtIriables of co""leteiy regrJtu growth, Tear. FUDkcii. Funkciona1. Anal. i PriloZen. Vyp. 30 (1978), 3-13. (Russian) AIm!zJnt, N. I. [1] On the theory of entire junctions of finite degree, Dokl. Akad. Nauk SSSR 6.1 (1948),

475-478. (Russian) [1] On aome properties of integral tTtl1l8Ce1fliental juncti01lS of exponentioI type, Izv. Akad. Nauk SSSR Ser. Mat. 10 (1946), 411-428. (Russian; English summary) [3] 0" the poIynomiola of B. M. Leviton, C. R. (Dokl.) Acad. Sci. URSS 54 (1946), 3-5. [4] 0" entire tl'OllSCendental junctions of finite degree htwing a mqjorant on a aequence of real pointa, Izv. Akad. Nauk SSSR Ser. Mat. 16 (1952), 353-364. (Russian) [5] Lectures on the theory of approximation, OGIZ, Mosc:ow, 1947; English transl., Ungar. New York, 1956. AIm!zJnt, N. I., KIuiiN, M. G. [1] Some flIII!8ti01lS i" the theory of momenta, Gos. NauCno.-Tebn. Izdat. Ukrain., Kbarkov,

1938; English transl., Amer. Math. Soc., Providence, R. I., 1962. AIm!zJnt, N. I., LIMN, B. JA[1] 0" interpolation of entire tramcendental juncti01lS of finite degree, Har'kov. Gos. Univ.

ui:en. Zap. 40 - Zap. Mat. Otd. Fiz.-Mat. Fait. i Har'kov. Mat. ObSC. (4) 13 (1952), 5-26. (Russian) AHn!zInt, N. I., MARi:l!NKo, V. A [1] On 8OI1Ie questions of approximations of continuow juncti01lS on the whole real axU. II, Har'kov. Gos. Univ. ui:en. Zap. 29 - Zap. Mat. Otd. Fiz.-Mat. Fait. i Har'kov. Mat. ObSC. (4) 11 (1949), 5-9. (Russian) AIILFoas, L. [1] Ei" StItz 00II Henri Carlan und mile A_ndung auf die 77Jeorie der I1IeI'OIIIOIphe Frmlctionen, Soc. Sci. Fenn. Comment. Phys.-Math. 5 (1931), DO. 16.

508

BIBLIOGRAPHY

509

AHLPOIlS, L., HEINs, M. (I) Quations of regularity connected with the Phrag",e,.-Lindelof principle, Ann. of Math. (2) 50 (1949),341-346. M. (I) Sur Ie dltplocement des zeros des fonetions entieres par leur derivation, These, Uppsala, 1914.

A.l.ANDER,

AMB~~,

V. (AMBARCUMJAN, V. A.) [I) Uber eine Frage der Eigenwerttheorie, Z. Phys. 53 (1929), 690-695.

ANDEIlSON, J. M. (I) Integral functions and Tauberian theorems, Duke Math. J. 31 (1965), 597-606. ANDWKO, M. I. [I) The extremal indicator of an entire junction of order less than one with positive zeros, Dopovidi Akad. Nauk Ukram. RSR 1_, 869-872. (Ukrainian) AzAIuN, V. S. [I) On rays of completely regular growth of an entire junction, Mat. Sb. 79 (Ill) (1969), 463-476; English transl. in Math. USSR Sb. 8 (1969). [1) A characteristic property of junctions of completely regular growth in the interior of an angle, Teor. Funkcii, Funkcional. Anal. i PriloZen. Vyp. 2 (1966),55-66. (Russian) [3) Generalization of a theorem of Hayman on subharmonic junctions in an n-dimensional cone, Mat. Sb. 66 (101) (1965), 248-264; English transl. in Amer. Math. Soc. Transl. (2) 80 (1969). (4) Functions of regular growth that are subharmonic in the whale space, Har'kov. 005. Univ. Ueen. Zap. 11.0 - Zap. Meh.-Mat. Fak. i Har'kov. Mat. ObSC. (4) :zs (1961), 128-148. (Russian) [5) On the indicator of a junction subharmonic in a multidimensional space, Mat. Sb. 51 (100) (1962),87-94. (Russian) [6) On the indicator of a function subharmonic in a multidimensional space, Dold. ADd. Nauk SSSR 139 (1961), 1033-1036; English transl. in Soviet Math. Dold.l (1961). [7) Subharmanic junctions of completely regular growth in multidimensional space, Dold. Akad. Nauk SSSR 146 (1962),743-746; English transl. in Soviet Math. Dold. 3 (1962). (8) Regularity of growth of entire junctions, Dold. Akad. Nauk SSSR lOG (1971), 511-512; English transl. in Soviet Math. Dold. Il (1971). (9) Extremal problems on entire junctions, Teor. Funkcii, Funkcional. Anal. i Priloien. Vyp. 18 (1973), 18-SO. (Russian) (10) The regularity of the growth of the Fourier coefficients of the logarithm of the modulus of an entire junction, Teor. Funkcii, Funkcional. Anal. i Priloien. Vyp. 27 (1977), 9-22. (Russian) (11) Asymptotic behavior of subharmonic and entire junctions, Dold. Akad Nauk SSSR 119 (1976),1289-1291; English transl. in Soviet Math. Dold. 17 (1976). (Il) Example of an entire junction with given indicator and lower indicator, Mat. Sb. . , (131) (1972),541-557; English transl. in Math. USSR Sb. II (1972).

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BERNSTEIN,

r

BERNSTEIN, V. [1) SlIT les proprietes caracteristiques des indicatrices de croissance, C. R. Acad. Sci. Paris 10l (1936), 108-110. (1) Sopra uno proposizione relatifXl alia crescenza delle funzioni %morfe, Ann. Scuola Norm. Sup. Pisa (2) 1 (1933),381-399. (3) Sulle proprieta caratteristiche delle indicatrici di crescenza delle trDlUcendenti were d' ordine finito, Reale Accad. Ital. Mem. CI. Sci. Fis. Mat Nat 7 (1936), 131-189. BoAS, JIL, R. P.

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tier Dif-

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CAaLsoN,F.

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8eqIIetIU

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BIBLIOGRAPHY GEL'PANO. I. M .• LEVITAN. B. M.

[1) On the determination of a differential equation from its spectral junction. Izv. Akad. Nauk SSSR Ser. Mat. 15 (1951). 309-350; English trans\. in Amer. Math. Soc. Trans\. (2) 1 (1955).

GEL'PAND. I.

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ABCDEFGHIJ -CM-89876S43210

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