An Introduction to Nonharmonic Fourier Series (Pure and Applied Mathematics (Academic Pr))

An Introduction to Nonharmonic Four ier Series

Pure and Applied Mathematics A Series of Monographs and Textbooks

Editors

Samuel Eilenberg and Hyman Bass Columbia University, New 'fork

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An Introduction to Nonharmonic Fourier Series

ROBERT M. YOUNG Department of Mathematics Oberlin College Oberlin, Ohio

1980

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Library of Congress Cataloging in Publication Data Young, Robert M An introduction to nonharmonic Fourier series

(Pure and applied mathematics) Bibliography: p. Includes index. 1. lourier series. 1. Title. 11. Series: Pure and applied mathematics, a series of monographs and textbooks. QA3.P8 (QA404l 510s (.51.5’.2433] 79-6807 ISBN 0- 12-772850-3

PRINTED IN THE UNITED STATES OF AMERICA 80 81 82 83

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CONTENTS

ix

Preface

1.

Bases in Banach Spaces

1 2 3 4 5 6 7 8 9 10

2.

Schauder Bases Schauder’s Basis for C[a,b] Orthonormal Bases in Hilbert Space The Reproducing Kernel Complete Sequences The Coefficient Functionals Duality Reisz Bases The Stability of Bases in Banach Spaces The Stability of Orthonormal Bases in Hilbert Space

1

3 6 15 19 22 27 30 37 41

Entire Functions of Exponential Type

Part One. The Classical Factorization Theorems 1 2 3 4 5

Weierstrass’s Factorization Theorem Jensen’s Formula Functions of Finite Order Estimates for Canonical Products Hadamard’s Factorization Theorem

54 59 62 69 74

Part Two. Restrictions along a Line 1 The “Phragmen-Lindelof” Method vii

80

Contents

viii

2 3 4 5

Carleman’s Formula Integrability on a Line The Paley-Wiener Theorem The Paley-Wiener Space

87 93 100 105

3. The Completeness of Sets of Complex Exponentials

The Trigonometric System Exponentials Close to the Trigonometric System A Counterexample Some Intrinsic Properties of Sets of Complex Exponentials 5 Stability 6 Density and the Completeness Radius

1 2 3 4

4.

112 118 122 126 131 137

Interpolation and Bases in Hilbert Space

Moment Sequences in Hilbert Space Bessel Sequences and Riesz-Fischer Sequences Applications to Systems of Complex Exponentials The Moment Space and Its Relation to Equivalent Sequences Interpolation in the Paley-Wiener Space: Functions of Sine Type Interpolation in the Paley-Wiener Space : Stability The Theory of Frames The Stability of Nonharmonic Fourier Series Pointwise Convergence

146 153 162 167 170 179 184 190 197

Notes and Comments

203

References

225

List of Special Symbols Author Index Subject Index

.

235 237 24 1

The theory of nonharmonic Fourier series is concerned with thecompleteness and expansion properties of sets of complex exponentials {eian'}in Lp[- n,n]. Its origins, which are classical in spirit, lie in the celebrated works of Paley and Wiener [ 19341 and Levinson [1940]. In recent years, in response to the development of functional analysis and, in particular, to the growing interest in bases in Banach spaces, research in the area has flourished. New approaches to old problems have led to important advances in the theory. This book is an account of both the classical and the modern theories. Its underlying theme is the elegant interplay among the various parts of analysis. The catalyst in the present case is the Fourier transform, through which the classical Banach spaces are mapped into spaces of entire functions. In this way, problems in one domain can be examined via their transform image in the other. The book is designed primarily for the graduate student or mathematician who is approaching the subject for the first time. Its aim as such is to provide a unified and self-contained introduction to a multifaceted field, not an exhaustive account of all that is known. Accordingly, the first half of the book presents an elementary introduction to the theory of bases in Banach spaces and the theory of entire functions of exponential type. At the same time, an extensive set of notes touches on more advanced topics, indicates directions in which the theory can be extended, and should prove useful to both specialists and nonspecialists alike. Much of the material appears in book form for the first time. The only prerequisites are a working knowledge of real and complex analysis, together with the elements of functional analysis. By that I mean roughly what is contained in Rudin [1966]. On occasion, when more advanced tools of analysis are required, appropriate references are given. Apart from this, the work is essentially self-contained, and it can serve as a textbook for a course at the second- or third-year graduate level. The problems, which are of varying degrees of difficulty, are an integral part of the text. Some are routine applications of the theory, while others ix

X

Preface

are important ancillary results- these are usually accompanied by an indication of the solution and an appropriate reference to the literature. A word about notation: Theorem 2.3 refers to Theorem 3 of Chapter 2. The labeling of all other results is self-explanatory. I am deeply indebted to Doug Dickson and Paul Muhly for their careful reading of the manuscript and for their sharp criticism and advice. I owe immeasurable thanks to Linda Miller, who proofread the entire book more times than I could possibly have hoped.

ROBERTM. YOUNG

1

1

BASES IN BANACH SPACES

Schauder Bases

Let X be an infinite-dimensional Banach space over the field of real o r complex numbers. When viewed as a vector space, Xis known to possess a Hamel basis-a linearly independent subset of X that spans the entire space. Unfortunately, such bases cannot in general be constructed, their very existence depending on the axiom of choice, and their usefulness is therefore severely limited. Of far greater importance and applicability in analysis is the notion of a basis first introduced by Schauder [1927]. Definition. A sequence of’ vectors { x , , x2, x 3 , . . .} in an injinite-dimensional Banach space X is said to be a Schauder basis for X if to each vector x in the space there corresponds a unique sequence of scalars { c l , c 2 , c 3 , . . . } such that

The convergence of the series is understood to he with respect to the strong (norm) topology of’ X ; in other words,

Henceforth, the term basis for an infinite-dimensional Banach space will always mean a Schauder basis. Example. The Banach space I p (1 5 p < a)consists, by definition, of all infinite sequences of scalars c = { c l ,c2, c3, . . .} such that IIcllp = (I l l c; n (= P)lip < m. The vector operations are coordinatewise. In each 1

2

Bases in Banach Spaces

[Ch. 1

of these spaces, the "natural basis" { e l , e 2 ,e 3 , . . .}, where e , = ( O , O , . . . , 0 , 1 , 0 , . . . ), and the 1 appears in the nth position, is easily seen to be a Schauder basis. If c = (c,). is in I", then the obvious expansion c = c,e, is valid. It is clear that a Banach space with a basis must be separable. Reason: If {x,} is a basis for X , then the set of all finite linear combinations c,.~,, where the c, are rational scalars, is countable and dense in X . It follows, for example, that since I" is not separable, it cannot possess a basis. The "basis problem"-whether or not every separable Banach space has a basis-was raised by Banach [I9321 and remained until recently one of the outstanding unsolved problems of functional analysis. The question was finally settled by Per Enflo [1973], who constructed an example of a separable Banach space having no basis. The negative answer to the basis problem is perhaps surprising in light of the fact that bases are now known for almost all the familiar examples of infinite-dimensional separable Banach spaces.

1

Problems 1.

2.

3. 4. 5.

6.

7. 8.

Prove that every vector space has a Hamel basis. Prove that every Hamel basis for a given vector space has the same number of elements. This number is called the (linear) dimension of the space. Show that a Hamel basis for an infinite-dimensional Banach space is uncountable. Show that the dimension of I"' is equal to c. (Hinr: Show that the set ((1, r, r 2 , . . .) : 0 < r < 1) is linearly independent.) Let X be an infinite-dimensional Banach space. (a) Prove that dim X 2 c. (Hint: Show that there is a vector space isomorphism between I " and a subspace of X . ) (b) Prove that if X is separable, then dim X = c. The Banach space c, consists of all infinite sequences of scalars which converge to zero (with the I" norm). Show that the natural basis is a Schauder basis for c,, . Exhibit a Schauder basis for the Banach space c consisting of all convergent sequences of scalars (with the I O U norm). An infinite series c x n in a Banach space X is said to be unconditionally convergent if every arrangement of its terms converges to the same element. It is said to be absolutely convergent if the series ~ I I . x , l l is

Sec. 2]

3

Schauder’s Basis for C[a.b]

convergent. Show that every absolutely convergent series in Xis unconditionally convergent. What about the converse? 9. A basis (x,} for a Banach space X is said to be unconditional (absolute) if every convergent series of the form ~ c , x ,is unconditionally (absolutely) convergent. (a) Show that the natural basis is unconditional for the spaces I ” , I S p < a,and co. Show also that it is absolute for I P only when p = 1. Is it absolute for co ‘? (b) Show that the sequence of vectors ( I , O , O , O , . . .), (1, l,O,O,. . .), (1, 1, l , O , . . .), . . . forms a basis for co which is not unconditional.

2

Schauder’s Basis for C[a,b]

One of the most important and widely studied classical Banach spaces is C[a, h ] , the space of all continuous functions on the closed finite interval [a, h ] , together with the norm

The celebrated Weierstrass approximation theorem asserts that the polynomials are dense in C[a, b ] : if f is continuous on [a, b ] , then for every positive number I: there is a polynomial P such that the inequality

holds throughout the interval [u, b ] . For a given continuous function, a sequence of approximating polynomials can even be given explicitly. The most elegant representation is due to Bernstein. Let us suppose, for simplicity, that f is continuous on the interval [0, 11. Then the nth Brrnstein polynomial for f is

As is well known, f ( x ) = lim B,(x) n + ,XI

uniformly on [0, 13 (see Akhiezer [1956, p. 301).

4

Bases in Banach Spaces

[Ch. 1

Since every polynomial can be uniformly approximated on a closed interval by a polynomial with rational coefficients, the preceding remarks show that the space C[a, b] is separable; in fact, it has a basis. The space C[a,b] possesses a basis.

Theorem I (Schauder).

Proof. We are going to construct a basis for C [ a , b ] consisting of piecewise-linear functions f.(n = 0, 1,2,. . .). This means that to each function f in the space there will correspond a unique sequence of scalars {c,} such that

c c,fn(x) UJ

f(x)=

n=O

unqorrnly on [a, b ] . Let {xo, xl, x2, . . .} be a countable dense subset of [a, b] with xo = a and x1 = b. Set

f o ( x )= 1

x-a

fl(x) = -. b-a

and

When n 2 2, the set of points {xo, xI,. . ., x,- 1. partitions [a, b] into disjoint open intervals, one of which contains x,; call it 1. Define

i

if if

0

f,(x) = 1

linear

x#l x = x, elsewhere

for n = 2, 3 , 4 , . . .. The sequence { f o , f l , f2,. . .} will be the required basis. For each function f in C[a, b] and each positive integer n , we denote by L,f the polygonal function that agrees with 1' at each of the points xo, xl,. . ., x, ; we denote by L,f the function whose constant value is f ( x o ) . Since f is uniformly continuous on [a, b ] , a simple continuity argument shows that

L,f

-

uniformly on

f

[a, b ] .

Therefore, we can write cu

f

=

Lof

+

c (Lnf

-JL1.f).

n= 1

We are going to show that there are scalars cl, c 2 , c 3 , . . . such that L , j - L , _ 1.f

= C,f,

( n = 1, 2,

3 , . . .).

Sec. 2]

5

Schauder‘s Basis for C[a.b]

For this purpose, we shall define a sequence of functions {go, gl, g 2 , . . .} recursively by the equations

The claim is that g,

=

L,f, whence

Since gn is a polygonal function whose only possible corners are at the points xn, x l , . . ., x,, it is sufficient to show that g,, agrees with f at each of these points. This is trivial for n = 0, and we proceed by induction. Since .f,(x,) = 1, it follows that g,(x,) = f(x,,); if i < n, then f,(xi) = 0, and it follows from the definition of g,,, together with the induction hypot hesis, that

This establishes the claim. Accordingly, every function f~ C[a, b] has at least one representation of the form

and we have only to show that this representation is unique. Suppose then 0 a,f, and that some function g has two different representations, say h,J;,. If N is the smallest value of n for which a, # b,, then

c;=,,

for every x. Choose x = x N . Since j ” ( x N ) = 0 whenever n > N , it follows that uN = h,. But this contradicts the choice of N , and hence a, = b, for every n . I

ProbIems 1.

Give a probabilistic interpretation of the Bernstein polynomials (see Feller [1966, Chap. VII]).

6

Bases in Banach Spaces

[Ch. 1

2.

Prove that the space C [ a ,b] is separable by showing that every continuous function on [a,b] can be uniformly approximated by polynomials with rational coefficients. 3. Let f be a continuous function on ( - GO, GO).Prove that if there is a sequence of polynomials { P I , P,, P,, . . .} such that Pn+ f uniformly on (-co, GO),then f must itself be a polynomial. 4. Let f be a continuous function on [ a , b]. Show that there is a sequence of polynomials {PI,P,, P , , . . .} such that f = Pnand the series converges absolutely and uniformly on [ a , b ] .

3 Orthonormal Bases in Hilbert Space In a separable Hilbert space?, a distinguished role is played by those Schauder bases that are orlhonormal- the basis vectors are mutually perpendicular and each has unit length. An equivalent characterization of such bases is that they are complete orthonormal sequences. (Recall that a sequence of vectors { fl,f2,f3,. . . } in a Hilbert space is said to be complete if the zero vector alone is perpendicular to every f , .) It follows readily from this characterization that every separable Hilbert space has an orthonormal basis. The most important property of an orthonormal basis (as opposed to any other basis) is the simplicity of all basis expansions. If {e, , e, , e3, . . .} is an orthonormal basis for a Hilbert space H , then for every element j’ E H we have the Fourier expansion

The inner product (1; en) is called the nth Fourier coefficient of f’ (relative to { e n } ) .When the Pythagorean formula is applied to this series, the result is Parseval’s identity :

The validity of Parseval’s identity for every vector in the space is both necessary and sufficient for an orthonormal sequence to be a basis.

t All Hilbert spaces are assumed to be infinite-dimensional.

Sec. 33

7

Orthonormal Bases in Hilbert Space

Since the linear transformation

from H into 1' preserves norms, it must also preserve inner products. Thus

for every pair of vectors f and 9 ; this is the generalized Parseval identity. Even if an orthonormal sequence { r, ). is incomplete, Bessel's inequality is always valid:

whenever 1' E H . This shows, in particular, that the Fourier coefficients of each element of H form a square-summable sequence. The Riesz-Fischer theorem shows, conversely, that every square-summable sequence is obtained in this way: if lcn12 < sx,then there exists an element f in H for which

(1;en)= cnr

n = I , 2, 3 , . . ..

The proof is trivial: simply choose f = I ; c,e,. ;= We conclude that if ! c , ~ )is a cornplote orthonormal sequence in H , then the correspondence .I' [ ( j ;o,,)) between H and lz is a Hilbert space isomorphism. I t follows that from a geometric point of view, all separable Hilbert spaces are "indistinguishable", that is to say, isomorphic. +

Example 1.

In l 2 the "natural basis" is orthonormal.

Example 2.

In L 2 [- n, n ] ,with the inner product

the complex trigonometric system { e'"'}? constitutes an orthonormal basis. That the system is orthonormal is obvious; we prove that it is complete.

8

Theorem 2.

Bases in Banach Spaces

[Ch. 1

The trigonometric system is complete in L2 [ - n, n]

Proof. The proof will establish even more. Suppose that

for some integrable function f defined on [ - n, n] and n = 0, f 1, f2, . It is to be shown that f = 0 a.e. Set

for t E [ - n, n]. Integration by parts shows that (g(r) - c)

for every constant c and n for n = 0 also, and put

=

dt

=

0

f 1, & 2, f3 , . . .. Choose c so that this holds

F(t) = g(t) - C .

Then F is continuous on [ -n, n] and F(n)= F( -n). Weierstrass’s theorem on approximation by trigonometric polynomials guarantees that for each E > 0 there is a finite trigonometric sum

such that l F ( t ) - T(t)l < E

It follows that

6 -E j 2n

n -n

whenever ( t l 5 n.

Sec. 3]

Orthonormal Bases in Hilbert Space

9

so that

Since E was arbitrary, F

=

0, so that g = c and f

=0

a.e. I

Consequently, every function f in L z [ - n,n] has a unique Fourier series expansion

(in the meant). Here f ( n ) denotes the nth Fourier coefficient of f relative to {e'"'},i.e.,

's'

f ( n ) = -2n

f'(t)e-'"'dt

(n = 0, f 1, f 2 , . . .).

-n.

By Parseval's formula,

The mapping f + { f ( n ) }is a Hilbert space isomorphism between L2[-n,n] and 1'. There is a simple but useful extension of Parseval's identity that is worth mentioning. If f E L2[ -n, n], let f be the Fourier transform of f : I

Proposition 1.

rn

For every function f E L2[ - n, n] and every real number A,

Proof. Put y(r) =

. f ' ( t ) e - ' A ' . Then

f(n

+ A ) = d(n)

t Pointwise convergence is of course much harder. A deep result of Carleson [ 19661says that the Fourier series of an 1.' function converges (to thc function) pointwise almost everywhere.

10

Bases in Banach Spaces

for every integer n. Since A is real, 1 f from Parseval's identity applied to g. I

1 = 1 y 11,

[Ch. 1

and the result follows

As an illustration, let us choose f to be the constant function 1. A simple calculation shows that

for all real x. Setting A = t / x , where t is real and not an integral multiple of n, we obtain the important identity

1 = sin2 r

Tc

1

c (nn + t ) 2 '

~m

Example 3. The space H2 (named after Hardy) consists of all functions .f analytic in the open unit disk (in the complex plane) whose Taylor coefficients are square-summable, i.e., f(z)

=

c

cnzn,

1

with

<

00.

n=O

n=O

The inner product of two functions f(z) in H2 is, by definition,

unznand g ( z ) =

=

c."=hnzn

m

It is clear that H2 can be identified with the (closed) subspace of L 2 [-n, n] spanned by the functions ein*with n 2 0. Let en(z) = z" for IzI < 1 (n = 0, 1,2,. . .); then the en's form an orthonormal basis for H2. The natural mapping

c m

(co, CL, c 2 , . . .)'

c,zn

n=O

between l2 and H2 is a Hilbert space isomorphism. Example4.

The space A' consists of all functions f that are analytic in

Sec. 31

Orthonormal Bases in Hilbert Space

11

the open unit disk and have finite area norm

Under pointwise addition and scalar multiplication, A2 is a vector space; when endowed with the inner product

it becomes a Hilbert space (see Problem 16).

Assertion : If

then the en's form an orthonormal basis for A*. The orthonormal part is simple: setting z = re", we have

-

2Jn+1Jm+1. onm = n+m+2

As for completeness, it must be shown that if f E A' and ( f , e n ) = 0 for n 2 0, then j ' = 0. This we accomplish by proving that if the Taylor series of ,j' is

then

12

Bases in Banach Spaces

[Ch. 1

Argue as follows :

Since the series is uniformly convergent in each disk term-by-term integration is permissible and yields

ss

IzI

m i m i 2

Otr

f ( z ) . F m d x d y = 271

2

c,dn,

n=O

n+m+2

6r

r2m i 2

= RC,-

m

+ 1'

Since I f ( z ) Z m l is integrable over {z:IzI < 13, ( f , zm) = lim r- 1

ss

71Cm

f ( z ) F m d x dy = ___ m + 1'

14 6r

and the result follows. Let us remark that we now know that the Taylor series of a function 1' in A' converges to f in the topology of A'. This could not have been deduced solely from the fact that the Taylor series o f f is known to converge (to f )uniformly on each closed subset of the disk (see Problem 17). If f E A', with Taylor series .f(z) = c,z", then Parseval's formula shows that

Ilf1I2

=

=

c -n + 1 lCnl2

n=O

This provides an alternative description of A2 as the class of all functions f(z) = cnzn analytic in the open unit disk for which

c:=o

m

The inner product of two such functions this case defined to be

in agreement with the original definition.

2."=oa,z"

and

IF=,, hnzn is in

Sec. 3]

Orthonormal Bases in Hilbert Space

13

Problems Prove that every separable Hilbert space contains a complete orthonormal sequence. 2. Give an example of a nonorthogonal basis for a separable Hilbert space. 3. Let ( e l , .. ., be an orthonormal subset of a Hilbert space H and f an arbitrary element of H . Show that the minimum value of 1 f I ciei1 is attained when and only when ci = ( f , ei) for i = 1,. . ., n. e,, e 3 ,. . .$ be a complete orthonormal sequence in a Hilbert 4. Let space H and suppose that f l , J ; , f 3 , . .. are arbitrary vectors in H such that 1.

cy=

Show that the sequence { f l , j 2 ,13,.. .} is also complete in H . 5. Prove that an orthonormal basis for a separable Hilbert space is unconditional. (For the definition of an unconditional basis, see Problem 9 of Section 1.) 6. Let X be a Banach space and x and y elements of X . We say that x is orthogonal to y, and we write x I y, if

(a) Show by an example that x Iy need not imply that y I x. (b) Show that the relations x I y and x I z need not imply that x I(y z ) . (c) Show that if X is a Hilbert space, then x l y if and only if (u.v) = 0. (d) Is every complete orthogonal sequence in X a basis? 7. Let H be a nonseparable Hilbert space. (a) Using the Hausdorff Maximal Principle, show that H contains a complete orthonormal subset. (b) Let ( L > , : u E A ) be an orthonormal subset of H and f an arbitrary element of H . Show that ( f , e,) = 0 for all but countably many values of u . (c) Show that every complete orthonormal subset of H has the same number of elements. This number is called the (orthogonal or Hilbert space) dimension of H .

+

14

Bases in Banach Spaces

[Ch. 1

(d) Show that two Hilbert spaces are isomorphic if and only if they have the same dimension. (e) Show that for each cardinal number CI, there is a Hilbert space whose dimension is K . 8. Let X be the vector space of all finite linear combinations of functions where the parameter 1 is real. An of the form eiA' ( - cx) < t < a), inner product in X is defined by

When X is closed by means of the metric generated by this inner product, we obtain a certain Hilbert space B2 ( B is for Besicovitch). (a) Show that the continuum of elements eiA'forms a complete orthonormal subset of 8'. (b) Show that B2 contains the important class of Bohr almost periodic functions. These are obtained by adding to X the limits of sequences of functions in X that are uniformly convergent on the entire real line. 9. (a) Show that each of the systems { $sin nt : n = 1 , 2 , 3 , . . .) and { 1, $ cos nt : n = 1,2,3,. . .) forms an orthonormal basis for L2[0, n]. (b) Show that the system { 1, cos n t , sin(n - ) ) r : n = 1 , 2 , 3 , . . .} forms an orthonormal basis for L 2 [ -n, n]. 10. Prove that the sequence of functions { 1, ?I, x2,. . .) is complete in ,!,'[a, b] for every finite interval [a, b]. 11. Show that an orthonormal sequence { e , , e 2 , e 3 ,...) in a separable Hilbert space H is complete if and only if Parseval's identity

holds for every vector f belonging to a complete subset of H . 12. (Vitali) An orthonormal sequence (en} in L 2 [ a ,b ] is complete if and only if

for every x in [ a , b ] .

Sec. 41

15

The Reproducing Kernel

13. (Dalzell) An orthonormal sequence { e n } in L 2 [ a ,h ] is complete if

and only if

JIl 1 e,,(t)dr

11-

14.

1

dx

=

( h - u)2 2 .

~

Show that when the Grum-Schmidt process is applied to the functions I , x. .?, . . . in L 2 [ - I , I], the resulting (complete) orthonormal se$Pn(.x),where quence is J n

+

P,(.x) =

15.

16.

1

d"

-(x2 -

2 n ! dx"

1)"

( n = 0, 1 , 2 , . . .)

is the nth Legendre polynomial. Show that the space H2 can be described alternatively as the class of all functions .f' analytic in the open unit disk for which the integral means

remain bounded as r + 1 . (a) Show that if .f'€A2, then

(Hint: The value of an analytic function at the center of a disk is equal to its average value over the entire disk.)

(b) Prove that A' is a Hilbert space. in A', then f n + f uniformly on every closed 17. Show that if f n + f subset of 121 < 1. Show by an example that the converse is false.

4

The Reproducing Kernel

Most of the important examples of Hilbert spaces are function spaces. The special properties of the functions considered, such as analyticity, enrich the structure of the space, which in return supplies added information about the functions.

16

Bases in Banach Spaces

[Ch. 1

Let H be a Hilbert space whose elements are real or complex-valued functions defined on a set S. We shall call H a functional Hilbert space if for every element x E S the “point-evaluation’’ functional

on H is bounded. This means that there is a constant M, such that for all f E H we have If’(x)l 5 M , 1 f 11. By the Riesz representation theorem, every bounded linear functional on H arises from an inner product, and so if x E S, there is an element K , E H such that

for every f.

f ( x ) = (f, K , )

The function K on S x S, defined by

is called the kernel function or the reproducing kernel of H .

Example 1. The sequence space 1’ provides a trivial example of a functional Hilbert space. Here S is the set of natural numbers, and K is given by

( { e n }is

the “natural basis” for 12).

Example 2. Consider the space H2. For

c

I/?[ < 1 the function g defined by

m

g(2) =

pnzn

n=O

belongs to H2 and

This shows that “evaluation at is a bounded linear functional on H2. It also shows that g = K , ] , so that the reproducing kernel of H2 is given by K(z,w) =

c znwn

n=O

K is called the Szego kernel.

=I .

1 - 2E’

Sec. 41

17

The Reproducing Kernel

The next proposition shows that the reproducing kernel of a (separable) functional Hilbert space can always be described explicitly in terms of an orthonormal basis for the space.

Proposition 2. l / ( e l ,e2, e 3 , .. .) is an orthonormal basis f o r u functional Hilhert space H and lf K is the kernel ,function q/ H , then

Proof. Let x be a fixed but arbitrary element of S. If K , is expanded in a Fourier series relative to { e n ) , then

The result follows at once from Parseval's identity (generalized) since

n=

I

Example 3. Consider the space A'. That A' is a functional Hilbert space is an easy consequence of the following mean-value formula for analytic functions. I f f is analytic in the closed disk D, = { z : I z - zoI 5 r } , then

To prove the formula, observe that for every p 5 r ,

by Cauchy's integral formula. Multiplying both sides by p and then integrating (with respect to p ) from 0 to r, we obtain

which is the desired result.

Bases in Banach Spaces

18

[Ch. 1

Suppose now that f € A 2 and lzol < 1. Choose r small enough so that D, is completely contained within the open unit disk. When the CauchySchwarz inequality is applied to this mean-value formula, the result is

This shows that “evaluation at zo” is a bounded linear functional on A’. To determine the reproducing kernel of A’, apply Proposition 2. The functions en defined by e,,(z)= , / m z r i for IzI < 1 and n = 0, 1,2,. . . form an orthonormal basis for A’ (see Section 3), and hence

-

1

ll(1 - ZW)” for IzI < 1, IwI < 1. The kernel function K of A2 is called the Bergmun kernel. The following integral representation for a function j ’ in A’ is now immediate :

Problems 1.

2.

Is the space L2[0,11 a functional Hilbert space? Let { e n } be an orthonormal sequence in a functional Hilbert space H ,

Sec. 51

Complete Sequences

19

with reproducing kernel K . Show that { e n } is complete if and only if

for every x. 3. A function K defined on S x S is called a positive matrix if for each positive integer ii and each choice of points t , , . . ., r, from S the quadratic form

is positive detinite.

(a) Show that the reproducing kernel of a functional Hilbert space is a positive matrix. (b) Show that if K is a positive matrix, then there is a functional Hilbert space whose reproducing kernel is K . 4. Let P be the Hilbert space of all entire functions of the form

where cp E L 2 [ -n, n]. The inner product of two functions f and g in P is defined to be

Show that P is a functional Hilbert space and find its reproducing kernel. (P is the famous Puley- Wiener space; it will play an important role in the theory of nonharmonic Fourier series.)

5

Complete Sequences

A basis for a Banach space has the important property that every vector in the space can be approximated arbitrarily closely by finite combinations of the basis elements. Of course, a basis is much more. In many cases,

20

Bases in Banach Spaces

[Ch. 1

however, it is sufficient to know that a sequence of vectors, although not necessarily a basis, does nevertheless have this approximation property. Such sequences will be called complete. Definition. A sequence of vectors {xl, x2, x 3 , . . .} in a normed vector space X is said to be completet fi its linear span is dense in X , that is, if for each vector x and each E > 0 there is a finite linear combination c 1x, + ’ . . cnx, such that

+

IIX

-

(C’XI

+ . . . + c,xn)II <

E.

It is a direct and important consequence of the Hahn-Banach theorem that the completeness of a sequence of vectors {x,} in X is equivalent to the following condition : if p E X * (the topological dual of X ) and if p ( x , ) = 0 ( n = 1,2,3,. . .), then p = 0. When X is a Hilbert space, the Riesz representation theorem shows that all is well-the former definition (see Section 3) agrees with the present one. Example 1. If (X, Gi?, p) is a a-finite measure space and 1 5 p < 00, then the Riesz representation theorem shows that the dual of Lp(p)can be identified with Lq(p), where l / p l / q = 1 . It follows that a sequence { f , } of functions in Lp(p)will be complete provided the relations

+

with g E Lq(p),imply that g = 0 a.e. If p is a finite measure, then Lp(p)c L’(p) so that if the relations above imply that g = 0 a.e. whenever g E L1(p),then the sequence { f,} will be complete in Lp(p)for every p 2 1. Example 2. The trigonometric system {ein’}?mis complete in Lp[-n, n] for 1 S p < 00. Indeed, the proof of Theorem 2 showed that iff E L’ [ - n,n] and

then f = 0 a.e.

t The terminology of the subject is not uniform-the terms “closed”, “total”, and “l’undamental” are also used.

Sec. 51

21

Complete Sequences

Example 3. In C[a,h ] the sequence of powers { 1, t , t 2 , . . .} is complete by virtue of the Weierstrass theorem on polynomial approximation : the polynomials are dense in C [ a , b ] . It follows that if y is an element of C [ a , h ] for which all of the "moments"

l

(n = 0, 1,2,. . .)

t"g(t) dt

are zero, then y must be identically zero. Indeed, if we write

then p is a bounded linear functional on C[u,h] and p ( t " ) = 0 for every n. Since the powers o f t are complete, p = 0, and hence g must be identically zero. Example 4. The sequence of powers { 1, t , t 2 , . . .} is complete in LP[a,b] whenever [a, b] is an arbitrary finite interval and 1 5 p < co. This follows readily from Examples 1 and 3. For suppose .f E L' [a, h ] and

l

(n = 0, 1,2,. . .).

t"f(t)dt = 0

If we define

then F E C[u, b] and F(u) = F(h) = 0. Integration by parts shows that

for n = 1.2.3,. . .. Conclusion: F everywhere.

=

0 everywhere so that f'

=

0 almost

Problems 1.

Show that if ( r , ) is a complete orthonormal sequence in a Hilbert space, then {r,, - r , , , } is also complete. Is the result true if { e n } is merely assumed to be complete?

Bases in Banach Spaces

22

2.

3.

4. 5. 6.

7. 8.

9.

6

[Ch. 1

Let {x,} be a complete sequence in a normed vector space X . Show that if {y,} is complete in {x,} (that is, if each x, can be approximated arbitrarily closely by linear combinations of the y,), then {y,} is complete in X. Show that the sequence { 1, x2, x4,. . .} is complete in C[O, I]. (Give an “elementary” proof, that is, one that uses neither the Miintz-Szhsz theorem nor the Stone-Weierstrass theorem.) Show that if { f , , f 2 , f 3 , . . .} is a sequence in C[O, 11 that is complete in L2[0,11, then { 1, f , , f 2 , .. .} is complete in C[O, 11. Show that the trigonometric system {eint}Tm is not complete in L 2 [ - A , A ] if A > n. (Boas-Pollard) Let { f,,, f,, fz,. . . } be a complete orthonormal sequence in LZ[a,b]. Show that there exists a bounded measurable function g such that the sequence {gf,,g f 2 , gf3,. . .} is complete in L 2 [ a , b ] . ( H i n t : Any bounded measurable function g that is never zero and is such that fo/g $ L2[a,b ] will do.) 1 is complete in Show that the sequence ___ -x + 1 ’ x + 2’x + 3 ’ ’ ” LZ[O, 11. Let D be a bounded domain (an open connected set) in the complex plane and A(D) the space of functions f that are analytic in D and continuous in the closure of D. Define 11 f 1 = maxf If(z)l :z E D}. (a) Show that if D is the open unit disk, D = {z:IzI < l } , then the sequence { 1, z, z 2 , . . .} is complete in A(D). (b) Show that if D is multiply-connected, then the sequence { 1, z , z2,. . .} is not complete in A(D). Let T be the unit circle in the complex plane, T = { Z : ~ Z = [ l } , and C ( T ) the space of all continuous complex-valued functions ,f on T . Define Ilfll = max{lf(z)l:zE T } . Show that the polynomials in z are not dense in C ( T ) . (Give an example of a continuous function on T that is not the uniform limit of polynomials.)

’{

The Coefficient Functionals

If {xl, x2, xj, . . .} is a basis for a Banach space X , then every vector x in the space has a unique series expansion of the form

c CII

x =

11=

1

c,x,.

Sec.6]

23

The Coefficient Functionals

I t is clear that each coefficient c,, is a linear function of x . If we denote this linear function by j;,,then c,, = f,,(x). and we may write u,

x =

1 f,(X)X,. n= 1

The functionals f , ( n = I, 2,3,. . .) thus defined are called the coefficient functionals associated with the basis {x,,}. They play an essential role in many parts of Banach space theory; perhaps the most important statement about them is that they are continuous. Theorem 3. I f {xn) i s a b a s i s for a Banach space X and if { f,,}i s the ussociated sequence of coej’icient functionals, then each f,,E X * . Moreover, there e x i s t s a constant M such thut

Proof. Introduce the vector space Y consisting of those sequences of scalars {c,,} for which the series xp=I c,x,, is convergent in X . If {c,,}E Y , then the number

satisfies all the properties of a norm. We begin by showing that Y , endowed with this norm, is a Banach space isomorphic to X . Verification that Y is a Banach space is typical of standard completeness arguments, and since it poses no special difficulties, it is left as an exercise for the reader. To see that X and Y are isomorphic, argue as follows. The mapping T : Y X defined by --+

n= I

is obviously linear; since { x , , } is a basis, it is also one-to-one and onto.

Since

it follows that T is continuous, and the open mapping theorem then guarantees that T - ’ is also continuous. This proves that X and Y are isomorphic.

24

Bases in Banach Spaces

Suppose now that x Then for every n,

=

cp==l cnxnis a fixed but

[Ch. 1

arbitrary element of X.

This proves that each f , is continuous and that IIf,II 5 2IIT-'Il/llx,II. Choosing M = 2 1 T - 11, we have IIxn II . I f n

II 5 M

for every n. The remaining inequality is trivial:

Definition. Let {x,} be a basis for a Banach space X and let i f , } be the associated sequence of coeflcient functionals. For n = 1,2,3,. . . the nth partial sum operator S, is the linear operator on X defined by

Theorem 3 implies that each partial sum operator is bounded, and a closer examination of the proof reveals further that

As an application of this fact, we establish a simple and useful criterion for

determining when a complete sequence is a basis.

Theorem4 A complete sequence {x,} of nonzero vectors in a Banach space X is a basis for X if and only if there exists a constant M such that

whenever

c,,

. . ., c,

are arbitrary scalars and n 6 m.

Sec.61

25

The Coefficient Functionals

Proof. The necessity is easy: we need only choose M = sup,IIS,II, where S,, is the nth partial sum operator associated with { x , ) . Reason: If m 2 n, then

now take the norm of both sides. For the sufficiency, start with a fixed vector x and select scalars { c i n } l = ,, n = 1 , 2 , 3 , . . ., such that n

x = lim n+ui

C cinxi. i= I

(This is possible because {x,,) is complete.) For notational convenience, put ci,, = 0 for i > n. For fixed k and m > n > k we have by hypothesis

As m ,n + r y j , this last expression tends to zero. Since x k # 0, it follows that for each k the sequence { c , ~ } ; = ~is a Cauchy sequence, and therefore the limit

exists. It is now a routine matter to show that

The details are left to the reader. We complete the proof by showing that this representation is unique.

26

Bases in Banach Spaces

[Ch. 1

It suffices to show that if C k X k = 0 , then ck = 0 for every k . For fixed k and n 2 k we have once again by hypothesis

Letting n + 00, we find

1

CkXk

11 = 0, and hence ck = 0 for every k .

I

Problems 1.

2.

3. 4.

5.

6.

7.

8.

Show that the normed vector space Y defined in the proof of Theorem 3 is a Banach space. Let { x , } denote Schauder’s basis for C[a, b] and let { f,} be the associated sequence of coefficient functionals. Compute 1 f, 11. Use the criterion of Theorem 4 to show that Schauder’s system is in fact a basis for C[a, h ] . Let { x,} be a basis for a Banach space X and { S,} the corresponding sequence of partial sum operators. Show that 1 5 SUP, 1 S, 1 < 00. (Karlin) A basis { x , } for a Banach space X is said to be normal if I(x,II = = 1 for every n. Prove that in a Hilbert space every normal basis is orthonormal. (Hint: Show that if { x , } is normal, then x, Ix, whenever n # m (see Problem 6 , Section 3).) A sequence {x,} of nonzero vectors in a Banach space X is said to be an orthogonal system if for each n ( n = 1,2,3,. . .) the linear span of { x , , . . ., x , } is orthogonal to x , + , (see Problem 6, Section 3). (a) Show that an orthogonal system is linearly independent. (b) Show that a complete orthogonal system is a basis. A Banach space X is said to have the approximation property (in the sense of Grothendieck) if the identity operator on X can be approximated, uniformly on every compact subset of X , by operators of finite rank. (a) Prove that every Banach space with a basis has the approximation property. (b) Let X be a Banach space with a basis and let Y be an arbitrary Banach space. Prove that every compact linear transformation from X into Y is the limit (with respect to the norm topology) of operators of finite rank. Prove that the disk algebra A has the approximation property. (See Problem 7. In the notation of Problem 8, Section 5 , A = A(D), with D = { z : [ z [< l}.)

I f,I

Sec.71

7

27

Duality

Duality

Suppose that ( x , ~is) a basis for a Banach space X and that { ,f,} is its associated sequence of coefficient functionals. What can we say about the sequence ( j;,}?Surely not that it is a basis for X * . For if X * is nonseparable, then it contains no basis at all (example: X = 1 ' ) . Therefore, unless ( I n } is complete, the most we can hope for is that it be a basis for its closed linear span. The hope is justified. The closed linear span of a sequence { x , } of elements from a normed vector space X will be denoted by [x,].

Theorem 5. Let (x,} be a basis for a Banach space X and let { f n ) be the ussociuted sequence of' coelicient functionals. Then { f,} is a basis for [j , ] and the expansion

i s ilalid jbr e w r y f in [ j , ] . Proof. The proof is based on the fact that the adjoint of the nth partial sum operator S, is given by the formula

for every j ' E X * . The formula is valid because

for every x E X . It is to be shown that S,*f + f for every f E [f,]. Let us first assume that f is a finite linear combination of the f , , say I c i j ; . Then for every n 2 rn f =

i= 1

i= 1

and hence trivially S a j ' + j ' . If f is an arbitrary element of [ J , ] , then given

E

> 0, we can find

28

Bases in Banach Spaces

g = E L l c i j ) such that IIf - y It follows that for all n 2 rn

1

< c/(M

+ I), where M

=

[Ch. 1

sup,IIS,,(I < m.

so that once again S,*f+f: Thus every element of [ j , ] has at least one representation of the form f = c,,f,. Since f ( x , ) = c,, for every n, the representation is unique and { f,} is a basis for its closed linear span. I

cr=l

When X is a reflexive Banach space, { f , } is a fortiori complete. Theorem 6. I f ( x , } is a basis f o r a reflexive Banach spuce X , then the associated sequence of coeficient functionals { f.} is a basis for X * . Pro@. In view of Theorem 5, we need only establish that { f,) is complete in X*. Suppose then that p E X** and that p( f , ) = 0 for n = 1 , 2 , 3 , . . .. It is to be shown that p = 0. Let P be the canonical imbedding of X into X**, i.e.,

for f in X * and x in X . Since X is reflexive, it follows that p = P x for some . .. Since { x , } is a basis, we have

x ; accordingly, .f;,(x)= 0 for n = 1,2,3,. x = I ~ , ( x ) x= , 0. Thus p = 0. I

EzL

For the remainder of this section X will denote a fixed Hilbert space. Two sequences {x,} and {y,,) in X are said to be biorthogonal if

for every rn and n. The Hahn-Banach theorem shows that for a given sequence { x , } a biorthogonal sequence {y,). will exist if and only if {x,,} is minimal? (this means that each element of the sequence lies outside the closed linear span of the others). If this condition is fulfilled, then the biorthogonal sequence {y,} will be uniquely determined if and only if { x , ) is complete. t The terms (topo/o~iccd/y) independen/ and

(topo/oyicu//~,),/rc.c, are also used.

Sec. 71

29

Duality

Suppose now that X is separable and that {xfl) is a basis for X . The isomorphism between X and X * shows that to each coefficient functional jiI there corresponds an element y, E X such that f,(x) = (x, y,) for all x. Since ffl(xm) = 6,,,, we see that {xll} and {y,) are biorthogonal. Thus we have shown that every basis {x,) for a Hilbert space possesses a unique biorthogonal sequence {y,}. In terms of this biorthogonal pair, each vector x in the space can be uniquely represented in the form

Combining Theorems 5 and 6, we see that { Y , ~ } is also a basis for X and that, by duality,

Thus, in a Hilbert space the sequence biorthogonal to a basis i s itself u basis.

Problems Let (x,) be a basis for a Banach space X and let { f,,}be the associated sequence of coefficient functionals. Prove or disprove: if X * is separable, then { f i l l is complete (and hence a basis) in X * . 2. A sequence { X , ~ ) of elements of a Banach space X is said to have { f , ] , where EX*, as a biorthogonal sequence if f , ( x j ) = S i j for every i and j. (a) Show that (x,} has a biorthogonal sequence if and only if it is minimal. (b) Show that a biorthogonal sequence for [xln} is uniquely determined if and only if { x,} is complete in X . (c) Show that { x l l } is a basis for X if and only if it possesses a bif,(x)x, converges to x orthogonal sequence (1,)such that for every x in X . 3. (Karlin) Let {x,} be a sequence of elements of a Banach space X and let be biorthogonal to {x,,}. Prove that if { f , } is a basis for X * , then {xll) is a basis for X . 4. Show by an example that in a Hilbert space, a sequence biorthogonal to a complete sequence need not itself be complete. 1.

Bases in Banach Spaces

30

8

[Ch. 1

Riesz Bases

The simplest and perhaps the most obvious way of constructing new bases from old is through an isomorphism of the underlying space. Thus, if { x , } is a fixed but arbitrary basis for a Banach space X and if the bounded invertible operator? T transforms {x,} into { y , } , that is, if Tx,

=

for n

y,

=

1 , 2 , 3 , .. .,

then { y , } is also a basis for X . In fact, it is easy to see that { x , } and {y,} are equivalent in the following sense. Definition. Two bases { x , } and { y , } f o r a Banach space X are said t o be equivalent if m

m

1 c,x,

is convergent

1 c,y,

if and only if

n= I

is Convergent.

n= 1

The property of being linked by an isomorphism of the space completely characterizes equivalent bases.

Theorem 7 . Two bases { x , } and { y , } f o r a Banach space X are equivalent if’ and only if’ there exists a bounded invertible operator T : X + X such that Tx, = y , f o r every n. Proof. The sufficiency is trivial. Suppose then that {x,} and { y , } are cnxnrthen the series equivalent bases for X . If X E X , with x = Z:=, c,y, converges to an element Tx in X . The function T thus defined is clearly linear, one-to-one and onto, and Tx, = y , for every n . To show that T is a bounded invertible operator, we define functions T , by setting T,x = c , y , . Then

c:==l

c;=

Tx

=

lim T,,x n-au

for every x . Since each T , is bounded (by Theorem 3), it follows that T is bounded (by the Banach-Steinhaus theorem). The open mapping theorem guarantees that T is invertible, and the result follows. I t”1nvertible” means one-to-one and onto

Sec.81

31

Riesz Bases

Theorem 8. I n onul sequences.

(I

Hilhert spuce equiiiulent buses have equivalent biorthog-

Proof: Let { s , ] and (y,) be equivalent bases for a Hilbert space H and let [ jiI) and { ~ g 1, ~ be their respective biorthogonal sequences. Observe to begin with that a sequence biorthogonal to a basis is also a basis, so that [jiIi and {g,l; are themselves bases for H. Let T be a bounded invertible operator on H such that Tx, = y, ( n = 1 , 2 , 3 , . . .). Then the adjoint operator T* is also bounded and invertible. Assertion: T*gll = ,Ll ( n = I, 2,3,. . .). Indeed, for fixed n and all values of m we have

since (x,; is complete, the assertion follows. This proves that {glI) are equivalent bases for H . I

if,)

and

In a separable Hilbert space the most important bases are orthonormal. Second in importance are those bases that are equivalent to some orthonormal basis. They will be called Riesz bases, and they constitute the largest and most tractable class of bases known. Definition. A husis for a Hilbert space is u Riesz basis if it is equivalent to an orthonorind husis, thut is, if it is obtained from an orthonormal basis h y meuns 01' ( I bounded inoertiblr operator. A Riesz basis to say,

{jiI}for a Hilbert space is necessarily

bounded, that is

In fact, if {I;,) is obtained from the orthonormal basis { e n } by means of the bounded invertible operator T, then for every n

It follows readily from this that if { f , ) is a Riesz basis, then the sequence jill1} is also a Riesz basis. Reason: There exists a of unit vectors [j;#/Il

Bases in Banach Spaces

32

[Ch. 1

bounded invertible operator S for which Sen = en ~

II f n I

( n = 1,2,3,. . .);

therefore, the operator T S is bounded and invertible, and (TS)e, =

~

f"

I II

( n = 1,2,3,. . .).

fn

The next theorem provides a number of important characteristic properties of Riesz bases.

Theorem 9. Let H be a separable Hilbert space. Then the .following statements are equivalent. The sequence {f,}forms a Riesz basis for H . There is an equivalent? inner product on H , with respect to which (2) the sequence { f , } becomes an orthonormal basis for H. (3) The sequence { f n } is complete in H , and there exist positive constants A and B such that for an arbitrary positive integer n and arbitrary scalars c l , . . . , c, one has (1)

(4)

The sequence { j;,}is complete in H , and its Gram matrix

generates a hounded invertible operator on 12. ( 5 ) The sequence { f,} is complete in H and possesses a complete hiorthoyonul sequence {g,} such that

for every j ' in H.

t Two inner products are said to be equivalent if they generate equivalent norms

Sec. 81

33

Riesz Bases

Proof. ( I ) + (2): Since [ j ; , )is a Riesz basis for H , there exists a bounded invertible operator T that transforms { j;,; into some orthonormal basis [ e l , ) i.e., ,

T/;$= en

for n

=

I, 2,3,. . ..

Define a new inner product ( f , g ) ] on H by setting

and let

)I /I1

be the norm generated by this inner product. Then

for every , / in H , so that the new inner product is equivalent to the original one. Clearly,

for every i and , j . (2)*(3): Suppose that (1;~)~ is an equivalent inner product on H relative to which the sequence {,/,,). forms an orthonormal basis. From the relations

where rn and M are positive constants not depending on f , it follows that for arbitrary scalars c, , . . . , c, one has

Clearly, the sequence ; j ; , i is complete in H . ( 3 )* (1): Let ( e , , )be an arbitrary orthonormal basis for H . It follows by assumption that there exist bounded linear operators T and S such that Tc,, = j ; , and S j , , = e,, ( n = 1,2,3,. . .). Certainly, ST = I . Since { f , ) is complete, we also have TS = I . Hence T is invertible, and { j ; , }is a Riesz basis for H . (1) 3 (4): Let T be a bounded invertible operator on H that carries

34

Bases in Banach Spaces

[Ch. 1

some orthonormal basis { e n } into the basis {j,)..If A = (aij)denotes the matrix of the (invertible) operator T*T relative to {e,}, then aij = (T*Tej, ei)= (Tej, Te,) = ( , f j , ji).

Therefore, the Gram matrix of { f,} is the conjugate of A . (4) (3): Suppose that the Gram matrix of { 1,)generates a bounded invertible operator on 1’. If { e n } is an arbitrary orthonormal basis for H , then the transformation T : H -+ H , defined by

whenever {ci}E l z , is obviously linear, bounded, and invertible. A straightforward calculation shows that

so that, in addition, T is a positive operator. (Recall that a bounded linear operator T on a Hilbert space is said to be positive if (Tj; 1’) 2 0 for every vector ,f in the space.) Since T is positive, it has a (unique) positive square root (see Riesz and Nagy [1955, p. 2651); call it P . The equation above may then be put in the form

from which it follows at once that

(1) =. ( 5 ) : Let {g,} be the unique sequence in H biorthogonal to { J ; , ) . By Theorem 8, { g,} is also a Riesz basis for H . Since every vector f ‘ in H has the two biorthogonal expansions

the result follows immediately from the definition of a Riesz basis.

Sec. s]

35

Riesz Bases

( 5 ) + (1): Consider the linear transformation from H into l2 defined

by

The reader will verify without great difficulty that this mapping is closed. By the closed graph theorem it is continuous, and hence there exists a positive constant C for which

Similarly, there exists a positive constant D for which

Fix an arbitrary orthonormal basis {en) for H , and define operators S and T on the linear subspaces spanned by the sequences {f,,} and {g,,}, respectively, by setting

ciei

and

T

By virtue of the two inequalities above, we have

and

Since both sequences {f,)and {g,,} are complete, each of the operators S and T can be extended by continuity to a bounded linear operator on the entire space. I f f = aif, and g = b j g j are finite sums, a simple calculation shows that

1

36

Bases in Banach Spaces

[Ch. 1

by continuity, this holds for every pair of vectors f and g. We have, accordingly,

so that S*T = I . The existence of a right-inverse for S* implies that S* is onto, and hence that S is bounded from below (see Taylor [1958, p. 2341). Since the range of S is dense in H , we conclude that S is invertible. Thus the sequence { f,} forms a Riesz basis for H . I The class of Riesz bases is very large. It is extremely difficult to exhibit at least one bounded basis for a Hilbert space that is not equivalent to an orthonormal basis. We mention without proof the following example of such a basis in the space L 2 [ - n , n]; it was discovered by Babenko [1948]. --oo, with 0 < c( < +, is a bounded Example. The sequence {Itreint},"= basis for L 2 [ -71, n] that is not a Riesz basis. The appearance of simplicity is misleading; the example is exceedingly difficult.

Problems Prove that a sequence that is biorthogonal to a Riesz basis is also a Riesz basis. 2. Suppose that { f,} is a sequence of vectors in a Hilbert space H such that cl(f, < co whenever f E H . Show that the mapping f + { ( f ,f , ) } has a closed graph. 3. Let { f , } be a Riesz basis for a Hilbert space H and let {g,} be biorthogonal to { f n } . Theorem 9 guarantees that there exist positive constants A and B such that 1.

f,)I2

Show that one has the dual relation

4.

Let { f,,} be a basis for a Hilbert space H and let { g f f } be biorthogonal to { f n } . We shall call { f,} a Bessel basis if m

a,

n= 1

n= 1

1 c , f n is convergent only if

IcnI2 c co;

Sec. 9l

The Stability of Bases in Banach Spaces

37

we shall call [ J,) a Hilbert basis if m

u,

C

cnfn

is convergent if

n= 1

1 lcnI2 < co. n= I

(a) Show that if,} is a Riesz basis if and only if it is both a Bessel basis and a Hilbert basis. (b) Show that { f,} is a Bessel (Hilbert) basis if and only if { g n } is a Hilbert (Bessel) basis. (c) Show that { Jn} is a Bessel basis if and only if there exists a constant A > 0 such that

for arbitrary scalars c I , . . ., cn (n = 1, 2,3,. . .). (d) Show that { f,,} is a Hilbert basis if and only if there exists a constant B > 0 such that

for arbitrary scalars c l , . . ., c, (n = 1 , 2 , 3 , . . .). Let { j ; , }be a basis for a Hilbert space H and let {g,} be its biorthogonal basis. Prove that { f n } and {g.} are equivalent if and only if { J n }is a Riesz basis. 6. Let { j i I }be a basis for a Hilbert space H and let g, = A,f,, where 5.

Prove or disprove:

9

{fn}

and {g,,} are equivalent.

The Stability of Bases in Banach Spaces

Two mathematical objects that are in some sense “close” to each other often share common properties. In this and the remaining section of Chapter I, we will show that bases in Banach spaces form a stable class in the sense that sequences sufficiently close to bases are themselves bases. Part of the problem is to formulate broad and effective notions of

38

Bases in Banach Spaces

[Ch. 1

“close”. Let {x,} be a basis for a Banach space X and let {y,,} be a sequence of elements in X . When shall we say that {y,} is “close” to {x,}? Although there are many different criteria, they all have one element in common: each implies that the mapping x,

+

y,

for n = 1,2,3,. . .

can be extended to an isomorphism T from X onto X that is in some sense “close” to the identity operator 1. In this way, questions about the stability of bases can be reduced to questions about “small” perturbations of the identity operator. As we shall see, the operator approach provides a powerful tool in the solution of stability problems. The fundamental stability criterion, and historically the first, is due to Paley and Wiener [1934]. It is based on the elementary fact that a bounded linear operator T on a Banach space is invertible whenever

(This is one of those striking instances in which linear operators behave like ordinary numbers: if 11 - tl < 1, then surely r-’ exists.) Let { x , ) be a basis for a Banach space X , Theorem 10 (Paley-Wiener). and suppose that {y,} is a sequence of elements of X such that

for some constant 1, 0 5 1 < 1, and all choices of the scalars cl,. . ., c, ( n = 1,2,3,, . .). Then {y,} is a basis for X equivalent to { x , } .

I.”=,

Proof. It follows by assumption that the series c,,(x, - y,) is conc,x, is convergent. Define a mapping vergent whenever the series T : X + X by setting

I.“=,

Evidently T is linear and bounded and IIT1( 5 A < 1. Thus the operator I - T is invertible. Since ( I - T ) x , = y, for every n, the result follows. I Corollary. Let {x,} be a basis for a Banach space X and let { f,} be the

Sec. 91

39

The Stability of Bases in Banach Spaces

associatd seyuencx~of

co
tors in X jbr which

fl=

1

thcn [ y,, 1 is a busis j b r X equivalent to {x,,}. Prooj; Put , I= sum, then

1 x,, - y, )I . 1 f ’ , I / .

If x

=

cixi is an arbitrary finite

Since 0 2 2 < 1 , the result follows from Theorem 10. I The following theorem is now immediate.

Theorem 11 (Krein-Milman-Rutman). If’ {x,} is a basis f o r a Banach space X , then there exist numbers E,, > 0 with the following property: if {y,,} i s a sequence of’ vectors in X for which

then { y,, 1 is a basis for X equivalent to {x,,). Let { j i } be the sequence of coefficient functionals associated with the basis { x I l ) .By the corollary to Theorem 10, it suffices to choose E, small enough so that I ~ , ~< ~1.f I ,,~~ Proof.

Zr=

In the following two corollaries the strength of the Krein-MilmanRutman theorem is forcefully revealed.

Corollary 1. I f X is a Banach space with a basis, then every dense subset of X contains a basis. Corollary 2.

The spuce C[a, b] has a basis consisting entirely of polynomials.

I t is important for applications that the corollary to Theorem 10 be

Bases in Banach Spaces

40

[Ch. 1

strengthened. For this purpose, we shall make use of a well-known result concerning “compact perturbations” of the identity operator. If T is a compact operator defined on a Banach space X and if Ker(I - T) = {0}, then I - T is invertible. This is the Fredholm alternative (see Halmos [ 1967, Problem 1401); it lends support to the notion that a compact operator is in some sense “small”. Definition. A sequence {x,} of elements of a Bunach space X is said to be w-independent if the equality

n= 1

is possible only for c,

=0

( n = 1,2,3,. . .).

Theorem 12. Let { x , } be a basis for a Banach space X and let { f,} be the associated sequence of coeflcient functionals. If {y,} is complete in X and if

then {y,} is a basis for X equivalent to {x,}. Proof. Let us first show that the sequence {y,) is cu-independent. If it were not, then we could find scalars cl, c 2 ,cg, . . . (not all zero) such that cny, = 0. Choose N so large that IIx, - y,ll . 1 < 1. It follows by the corollary to Theorem 10 that the sequence of vectors

X;=,+,

f,I

forms a basis for X. Accordingly, at least one of the scalars c I, . . ., cN does not vanish; suppose ck # 0. The equation c,y, = 0 can then be solved for yk, showing that X = [y,] = [ y n l n f k .Since 1 5 k S N, it follows that the codimensiont of [yn],>N is at most N - 1. But this is absurd: the quotient space X / [ y n ] , > N is isomorphic to [xl, . . ., X N ] , and so has dimension N. This proves that {y,} is o-independent. Define an operator T on X by setting

cF=l

t If Y is a closed subspace of a Banach space X,then the codimension of Y is defined to be the dimension of X/Y .

Sec. 101

The Stability of Orthonormal Bases in Hilbert Space

41

It is clear that T is a bounded linear operator. We claim that T is comf i ( x ) ( x i- y i ) , then by hypothesis pact. Indeed, if we put T,x = /I T - T,I/ -0, so that T is the limit of operators of finite rank, and hence compact. We complete the proof by showing that Ker(I - T ) = ( 0 ) .If (I - T ) x = 0, then f,(x)y, = 0 and since { y , } is o-independent, it follows that x = 0. Thus the kernel of I - T is trivial. The proof is over: the Fredholm alternative shows that I - T is invertible and clearly (I - T ) x , = y, for every n. I

I:=,

Problems 1. Show that if I = 1, then Theorem 10 is no longer valid. 2. Suppose that {x,} is a complete sequence of vectors belonging to a Banach space X and that 1 ci(xi - yi)ll 5 Ill cixi1 for some constant I , 0 5 I < 1, and arbitrary scalars cl,. . .,c, (n = 1,2,3,. . .). Show that { y f l }is also complete. 3. (Retherford-Holub) Let {x,) be a basis for a Hilbert space H with ~ ~ x=l 1, for ~ ~every n , and let { y , } be biorthogonal to {xn}.Show that < a,then { x l l } is a Riesz basis. (Hint: Use if Theorem 12 to show that { y , } is equivalent to {x,}, and then use Problem 5 of Section 8.)

1

d

10

m

The Stability of Orthonormal Bases in Hilbert Space

Throughout this section H will denote a separable Hilbert space and { e f t }a fixed but arbitrary orthonormal basis for H . While it is of course true that every result of the preceding section applies equally well to orthonormal bases, the added structure of a Hilbert space provides additional and stronger stability criteria. We begin by reformulating Theorem 10. Theorem 13. Let ( e n }he an orthonormal basis for a Hilbert space H and let { be "close" to {en}in the sense that ff1}

I , 0 2 I < 1, and urbitrury scalars c l , . . .,c, ( n = 1,2,3,. . .). Then { , f ; , }is a Riesz busis .for H .

j o r some constunt

42

Bases in Banach Spaces

[Ch.1

Application : The stability of the trigonometric system

It follows from what has already been done that the trigonometric system { e i n f } ? mis stable in L 2 [ -n, n] under "sufficiently small" perturbations of the integers. This means that if {A,} is a sequence of real or complex numbers for which {A, - n } is in some sense "small", then the system {eiAnf})"C', will form a basis for L2[-n,n], in fact, a Riesz basis. Accordingly, every function f in L2[-n,n] will have a unique nonharmonic Fourier series expansion 'w

f (t)=

C c,eianf

(in the mean),

-m

with x l c n l 2 < co. The possibility of suc.1 nonharmonic expansions was I iscovered by Paley and Wiener [1934], and it was for this purpose that they formulated the criterion of Theorem 13. In the present setting that criterion takes the form

whenever x l c n 1 2 5 1. When shall the sequence {A, - n } be considered "small"? Based on what has already been established, one might well suppose that the condition

A,,-n+O

as n - + + c o

is, at the very least, necessary. Surprisingly, it is not.

Theorem 14 (Kadec's +-Theorem). Zf {A,,} is a sequence of real numbers for which

then { e i a n fsatisfies } the Paley-Wiener criterion and so forms a Riesz basis for L2[ - n,n]. Proof. It is to be shown that

Sec. 103

whenever

The Stability of Orthonormal Bases in Hilbert Space

43

XIC,,~~ 5 1 . Write

where ii,,= A,, - n ( n = 0, f 1 , f 2 , . . .). The trick is to expand the function 1 - ei6‘ ( - n 5 t 5 n ) in a Fourier series relative to the complete orthonormal system [ I , cos n t , sin(n - i)r);=I (see Problem 9, Section 3) and then exploit the fact that 1,111- nl is not too large. Simple calculations show that when 6 is real,

+ i

C (-l)k26cosnb s i n k - k)t. . n((k - 3)’ - 6 2 ) ,

k

1

Let (c,,) be an arbitrary finite sequence of scalars such that xIc,12 5 1. By interchanging the order of summation and then using the triangle inequality we see that

where

and

Obvious estimates show that

But the series

2L/n(k2 - L 2 ) and

C?=,2L/n((k - i)’- L 2 ) are the

Bases in Banach Spaces

44

[Ch.1

partial fraction expansions of the functions l/nL - cot 7cL and tan nL, respectively (see Markushevich [ 1965, pp. 62, 64]), so that

The proof is over: L < & implies 1 < 1. I Remarks,

a

1. The result is sharp in the sense that the constant cannot be improved. In fact, if L = then the conclusion of the theorem no longer holds. A counterexample is provided by the sequence (A,,}, with

a,

n-&,

n>0, n

n+t,

=

0,

n<0.

Writing it down is easy; proving that it works is another matter (and hard). We defer the proof until Section 3.3. 2. The proof of Theorem 14 applies only when every 1, is real. Even earlier, however, it had been shown by Duffin and Eachus [1942] that the Paley-Wiener criterion is satisfied whenever the A,, are complex and n

= 0,

+1, f 2 ,....

The method of proof is the same, only now the function 1 - ei" is expanded in an everywhere-convergent Taylor series (see Problem 2). Whether the constant (log2)ln can be replaced by (for complex A,) remains an unsolved problem.

a

Any further analysis of nonharmonic Fourier series in L2[-n, n] requires the use of deep structural properties of entire functions. These will be discussed in Chapter 2. At present we can go no further. Let us return then to the abstract setting. It follows at once from Theorem 13 that if

Sec. 101

The Stability of Orthonormal Bases in Hilbert Space

45

then (jil)is a basis for H equivalent to (e,).,that is, a Riesz basis. If (1,) is known to be to-independent, then more can be said. We first give a definition. Definition. Two sequences of urctors { 1,) and {y,} in a normed vector space ure suid to he quadratically close if

Theorem 15 (Bari). Let H he a separable Hilbert space and { e f t ) an orthonormal basis for H . I f { is an to-independent sequence that is quadratically closr to { e l l ; then , i f f l ) is a Riesz basis f o r H . Proof:

Define an operator T : H + H by setting

It is clear that T is linear and that

Furthermore, since Te,, = eft - f,,, it follows that

This shows that T is a Hilhert-Schmidt operator and hence compact (see Halmos [1967, Problem 1351). We complete the proof by showing that Ker(I - T ) = lo).. If ( I - T),f'= 0, then from the equations

n

n

n

and the fact that /;,) is (independent, it follows that f = 0. Hence Ker(l - T ) = (01, and the Fredholm alternative shows that I - T is invertible. Clearly ( I - T)e, = 1, for every n. 1

46

Bases in Banach Spaces

[Ch. 1

A basis that is quadratically close to an orthonormal basis is called a Bari basis. Bari bases form an important subclass of the class of all Riesz bases. An excellent discussion of some of their special properties, together with applications to the theory of non-selfadjoint operators, can be found in Gohberg and Krein [1969] (also see Problems 5-7 at the end of this section). Since every orthonormal sequence of vectors is w-independent, the following result is an immediate consequence of Theorem 15.

Theorem 16 (Birkhoff-Rota). Let (el,} be an orthonormal basis for a Hilhert space H . l j {jn}i s an orthonormal sequence that is yuudratically , { f,} is complete, and hence an orthonormal basis. close to { q I }then Remark. I t is possible to give an elementary proof of Theorem 16. In this context, a proof is “elementary” if it depends only on the intrinsic geometry of the Hilbert space and not on operator theory. The reader will appreciate the power of the operator approach by tackling the “elementary” proof sketched in the exercises (see Problem 4).

Application : Eigenfunction expansions

An important class of boundary-value problems of mathematical physics, including the classical vibration problems of continuum mechanics (vibrating strings, membranes, and elastic bars, for example) can be reduced to the (regular) Sturm-Liouville system u”

+ (A - q(t))u = 0

(1)

(y(t) is assumed to be of class C’ on [a, b ] ) , together with the separated

endpoint conditions

+

~ ( a ) u’(a) = 0 ,

bu(h)

+ u’(b)= 0.

(2)

The following facts about such a system are well known (see Birkhoff and Rota [ 19621). 1, There is a discrete set of real eigenvalues A = A,, (n = 0, 1,2,. . .), and ,Il, as n m ; 2. eigenfunctions associated with different eigenvalues are orthogonal in L 2 [ a ,h] ; and -+

-+

Sec. lo]

The Stability of Orthonormal Bases in Hilbert Space

47

3 . the corresponding normalized eigenfunctions are given by the asymptotic formulas U,,(f)=

( n = 0, 1 , 2 , . . .),

J2 cos

where 0 , J f ) is a bounded function of n and t . Accordingly, the solutions to the system above behave in much the same way as the solutions to the system u”

+ l u = 0,

with the endpoint conditions u’(u) = u’(h) = 0 . It is now a simple matter to show that the eigenfunctions associated with any Sturm-Liouville system of the form (I), (2) are always complete in L2[u,b ] .

Theorem 17. ff u,, ( n = 0, 1,2,. . .) denotes the nth normalized eigenjunction u j the sysfrm (I), ( 2 ) , then the u,,’s form an orthonormal basis for LZ[a, b ] . Proof. In L 2 [ a ,h ] , the functions en defined by e,(t) = 1 and e,(t) =

(n2f--aa))

,has

for a

t

5 b (n = 1 , 2 , 3 , . . .)

form an orthonormal basis. The asymptotic formula for u,, shows that

c

IIen

-

and the result follows at once from Theorem 16. I Consequently, every function .f in L Z [ a , h ]can be expressed in terms of the eigenfunctions ; the corresponding series

is called a Sturm Liouoille series.

48

Bases in Banach Spaces

[Ch. 1

We conclude our discussion of stability by further clarifying the relation between Riesz bases and the Paley-Wiener criterion. As always, H is a separable Hilbert space and {ell}an orthonormal basis for H . The Paley-Wiener criterion is nothing more than the assertion that the mapping T:e,+f,

for

n

=

1,2,3,..

can be extended to an isomorphism on all of H for which 111 - TI1 < 1. This is a stringent requirement to place on a linear operator, and one might well conclude that the Paley-Wiener theory is of very limited scope. In a sense, just the opposite is true: every Riesz basis for H is obtained in essentially this way. a Riesz busis Theorem 18. Let H be a separable Hilbert space and {f;,} for H. Then there exists an orthonormal busis { e l l } ,an isomorphism T , and u positive number p such that Ten = pf;,

for

n

=

1,2,3,. . .

and

The proof will require the following two well-known facts (see Riesz and Nagy [1955, p. 2301 for the first and Halmos [1967, Problem 1051 for the second). Lemma 1.

I f T is a bounded self-adjoint operator on a Hilbert space, then

Lemma 2 (Polar Decomposition). Every hounded invertible operutor T on u Hilbert space can be juctored in the form T = U P , where U is u unitury operator und P i s a positive operator.

Proof of Theorem 18. Since { . f n } is a Riesz basis for H , there exist positive constants A and B such that

Sec. 101

whenever

The Stability of Orthonormal Bases in Hilbert Space E 1'.

49

Choose

2

P=JA+p and set (I,, = f ~ . f ; ,( n = I,2,3,. . .). The inequalities above may then be put in the form

where

We complete the proof by showing that there exists an orthonormal basis [e,l; such that

Select an u r h i l m r y orthonormal basis S: H - H by putting

{4,1)for H and define a mapping

whenever (c,~ E 1'. Then S is a bounded linear operator and

for every .f in H . This shows, in particular, that S is bounded from below. Since (g,,; is complete. S has a dense range and so must be invertible. Let S = U P be the polar decomposition of S. Then

1 91'11 and by Lemma 1,

5 (I

+ A)1 .f' I(

whenever f E H ,

Bases in Banach Spaces

50

[Ch. 1

and again using Lemma 1, we find

Define en = U&,, (n = 1,2,3,. ..). Since U is unitary, the sequence { e n ) forms an orthonormal basis for H . If { c,,} is an arbitrary square-summable sequence of scalars and if we put f = c,&,, then

1

The proof is complete since the mapping T: all the desired properties. I

x:=

c,e,,

+

2:i

cl,y,, has

Problems 1. Let { e n }be an orthonormal basis for H and let { f i 1 } be a sequence of

vectors in H such that

~ ~00. Show that { f n } is complete. Must it be a whenever 0 < I I C , , < basis? 2. (DuffiwEachus) Show that the system of exponentials { eiAn‘}:=- rx, forms a Riesz basis for L 2 [ - x, n] whenever

(Hint: Show that the Paley-Wiener criterion is satisfied by expanding the function 1 - eiJ‘in an everywhere-convergent Taylor series and then mimicking the proof of Theorem 14.) 3. (Schafke) Let { e n } be an orthonormal basis for H and let { j , } be “close” to { e l l }in the sense that

Sec. 101

The Stability of Orthonormal Bases in Hilbett Space

Show that

4.

51

[!it) is a Riesz basis for H . ( H i n t : Define

c,:=

and then show that T*f = I ( j ;e,,)(e,- f,).) Alternutive proof of’ Theorem 16. (a) Choose N large enough so that lie,, - j;,1I2< 1 . Then the system

is complete in H . (b) Define

Then

is also complete in H . (c) Let S be the orthogonal complement of the set { f N + f N + 2 , f ~ 3+, . . .). Then S = [ g l , . . . ,g ~ ] . (d) Finally, S = [ f l , . . . , f N ] . Therefore, if f Ij ; , ( n = 1 , 2 , 3 , . . .), then j ’ = 0. 5. If { f , l } is a Bari basis for H , then so is { f , / l f,1 }. 6 . In order that a sequence of vectors { A , } be a Bari basis for H , it is necessary and sufficient that there exist an orthonormal basis {e,,} and an invertible operator T on H such that (1) Tr,, = j;, for n = 1 , 2 , 3,..., and (2) I - T is a Hilbert-Schmidt operator. 7. Let { j ; , }be a basis for H that is quadratically close to the orthonormal basis { e , , } and let (9,) be the basis biorthogonal to {f,,}.Show that {gl,; is also quadratically close to { e , , } and hence that the bases {f,} and {g,,} are quadratically close.

This Page Intentionally Left Blank

2

ENTIRE FUNCTIONS OF EXPONENTIAL TYPE

There is an intimate connection between analytic functions and the completeness of sets of complex exponentials lei"'}. If, for example, the set {eiAn'}fails to be complete in C[a,b], then by virtue of the Riesz representation theorem, there is a function w ( t ) of bounded variation on the interval [a, h ] that is not essentially a constant and for which

jUb

e"n'dw(t)

=

(n = 1, 2, 3 , . . .).

0

If we let f ( z ) be the Fourier-Stieltjes transform of w, i.e., if

then f ( z ) is an entire function, not identically zero, and f(z) vanishes at every A,,. In this way the study of the completeness properties of the system {eiLnf} reduces to the study of the zeros of certain entire functions. The representation above for an entire function f(z) places a severe restriction on its growth. On the real axis, f(z) evidently is bounded, while off the axis, it can grow no faster than an exponential:

An entire function (or any function analytic in a sector) satisfying such an inequality for suitable constants A and B is said to be of exponential type. Familiar examples include ez, sin z, cos as well as all polynomials. Any limitation on the growth of an entire function carries with it limitations on the distribution of its zeros. In fact, the fundamental question in the theory of entire functions is precisely that of the connection

&,

53

Entire Functions of Exponential Type

54

[Ch. 2

between growth and distribution of zeros. The more zeros a polynomial has, for example, the higher its degree and hence the more rapid its growth. The relation between the growth of an entire function and the distribution of its zeros was investigated by Borel, Hadamard, Lindelof, and others in the late nineteenth and early twentieth centuries. The basis for this investigation is the classical theorem of Weierstrass on the expansion of entire functions into infinite products. Accordingly, we begin with the Weierstrass factorization theorem.

PART ONE.

THE CLASSICAL FACTORIZATION THEOREMS

1 Weierstrass’s Factorization Theorem Every polynomial can be written as a product of linear factors. Entire functions behave in much the same way: if f(z) is entire and has only finitely many zeros, zl,. . . ,z, (we shall always assume that a zero of order k is repeated k times), then we can “factor out” the zeros and obtain

where g(z) is entire and neuer zero. Any attempt to extend this process to the case of infinitely many zeros encounters serious convergence problems ; a more subtle approach is therefore needed. Let z, ,z 2 , z 3 , . . . be an infinite sequence of complex numbers, none of which is zero, with lim z, = 00. We are going to construct an entire function with precisely these zeros. The most natural choice for such a function is the infinite product

ir -;).

n= 1

(1

Observe, however, that unless the series C;=, l/Iz,,l is convergent, the product cannot converge absolutely (except for z = 0) and therefore may not represent an entire function. What is needed, in general, are “convergence-producing” factors. We are going to show that there exist polynomials p,(z) such that the product

Pt. 1, Sec. 11

55

Weierstrass's Factorization Theorem

converges to an entire function uniformly on each bounded region of the plane, and that p,(z ) can be chosen to be ( n = 1,2,3,. . .).

For this purpose we introduce the Weierstrass primary factors E(u,O) = 1 - u

and E(u, p ) = ( 1 - u ) exp

for p = 1,2,3,. .

P

Note that when IuI < 1, log E(u,p) = -

C

uk -

k=p+l

(the principal value of the logarithm is to be chosen), so that when IuI 2

E

< 1,

It follows easily from this estimate that the series

converges uniformly on each bounded region that contains none of the points z,,. Accordingly, if p,(z) is given by (2), then the infinite product (1) converges uniformly on every bounded region of the plane and so represents an entire function with zeros at each z , and at these points only. It is now a simple matter to show that every entire function can be factored in this way. Theorem 1 (Weierstrass). Every entire function f ( z ) that is not identically zero cun be represented in the form

56

Entire Functions of Exponential Type

[Ch. 2

where the product is taken over all the zeros of f ( z ) other than z = 0, m is a nonnegative integer, g ( z ) i s an entire function, and the pn(z)are given by (2).

Proof. Let zl, z 2 , z 3 , . . . be the zeros o f f ( z )other than z = 0, and suppose that f ( z ) has a zero of order m at the origin. In the case of infinitely many zeros, it is clear that { z f l }can have no finite limit point, and hence lim z, = a. If we define

with p,(z) given by (2), then +(z) is entire and has the same zeros as f’(z). Therefore, f (z)/+(z)is entire and never zero, so that

for some entire function g(z). The result follows. I The usefulness of the Weierstrass factorization theorem is limited by the fact that the polynomials pn(z)are of such large degree. There is, however, one case in which the expansion of j ( z ) can be greatly simplified. Let us suppose that for some nonnegative integer p , “ 1 < clznIP+

00.

n= 1

The estimate (3) can once again be applied, and we conclude in the same way as before that the product P(z) =

fl . ( t ? P ) u)

fl=1

converges uniformly (and absolutely) on each bounded region of the plane. Therefore, in this case, f ( z ) can be written as f ( z ) = zmeg(z)P(z).

cL ;

If p is the smallest nonnegative integer for which the series l/lz,IP+ is convergent, then P(z) is called the canonical product associated with the sequence { z n } , and the number p is called the genus of the canonical prod-

Pt. 1, Sec. I]

57

Weierstrass's Factorization Theorem

uct. We shall show subsequently that a canonical factorization is always possible provided that f(z) is not of too rapid growth. Example. As an application, we obtain the canonical expansion of sin nz. Here there is a simple zero at each integer: zn = n ( n = 0,f 1, f2,. . .). Since l/n diverges while l/nz converges, the genus of the corresponding canonical product is p = I , and we have

I:=,

IF=,

To determine g(z), take the logarithmic derivative of both sides. Then 71 cot

nz

=

I -

Z

+ g'(z) + n

+

;),

( 2 -L n + ~

But n cot nz = -1

+

Z

c (Jn+O

+

;),

z-n

so that g'(z) = 0 and hence y(z) is a constant. Since lim,,n(sinnz)/z it follows that eg(" = n, and hence

=

n,

Now, the product converges absolutely for all values of z, so that rearrangement is permissible, and we obtain the simpler formula sin nz

=

nz

fi

(1 -

n= 1

s).

Problems 1.

Prove that for IzI < 1, (1

+ z ) ( ~+ z2)(i + z4)(1 + z8).-

=

~

1 1 -zz'

58

2.

3. 4. 5.

6.

7.

8. 9.

Entire Functions of Exponential Type

[Ch. 2

+

Show that when z,, = i/n ( n = 1 , 2 , 3 , . . .), the product n(l z,) diverges while the product rill z,( converges. Exhibit a sequence {zn} of real numbers such that z z , converges and n(l z,) diverges. Suppose that Clznl2 < 00. Show that the product n(l zn) is convergent if and only if the series z, is convergent. Show that the value of an absolutely convergent product remains the same if its factors are reordered. If the product n(l - z/z,) is absolutely convergent for all values of z, then it represents an entire function. What if it converges conditionally for at least one value of z? Find an infinite product expansion for each of the following entire funtions: (a) cos z, (b) cosh z, (c) sinh z, (d) eZ - 1 , (e) e""- ehz. n 224466 Derive Wallis's product: - = - -- - ~. . . . 2 133557 Show that

+

+

+

~

-

provided that a is not a multiple of n. 10. Show that every function that is meromorphic in the entire plane is 11.

12.

the quotient of two entire functions. Let z l , z2, z 3 , . . . be a sequence of distinct complex numbers with lim z, = 00 and let wl, w2, w 3 , . . . be arbitrary complex numbers. Then there exists an entire function f ( z ) such that f ( z , ) = w, ( n = 1 , 2 , 3 , . ..). ( H i n t : Show that if g(z) has a simple zero at each z,, then there exist constants c, ,c 2 ,c 3 , .. , such that the series

converges uniformly in every bounded region of the plane.) Let zl, z 2 , z 3 , .. . be a sequence of distinct complex numbers with limz,, = co and let w I , w2, w 3 , . . . be arbitrary complex numbers. Show that there exists an entire function f ( z ) such that

j:"

.f(z)dz = w,

( n = 1,2,3,. . .),

Pt. 1, Sec. 21

2

59

Jensen's Formula

Jensen's Formula

In this section we shall establish a relation between the modulus of an analytic function on a circle and the moduli of its zeros inside the circle. It is known as Jrnwn's formulu, and it is one of the most important theorems in analysis.

Theorem 2 (Jensen's Formula). Let f ( z ) he analytic in Iz < R and suppose that f ( O ) # 0. !/' z I , . . . , z , , ure the zeros qf f ( z ) in Iz 5 r (0 < r < R ) , then

Proof. The result is immediate when f ( z ) is never zero in IzI S r , for then log1 j'(z)l is harmonic and the mean-value property gives

The essence of the proof is the observation that this formula remains valid even when j ( z ) has zeros lying on the circle IzI = r (but not in its interior). Suppose then that zl,. . . ,z, all lie on IzI = r, and write zk = reiek ( k = 1, . . . , n). There is but one thing to d o in the presence of unwanted zeros: we eliminate them. Define ~~

Then y(z) is analytic and free of zeros in IzI we obtain

r . Replacing f by g in (l),

Using the calculus of residues, it is not hard to show that

60

Entire Functions of Exponential Type

[Ch. 2

(see Ahlfors [1979, p. 160]), and this implies

lo2'

log11 - eiol dB = 0.

It follows that each of the integrals on the right-hand side of (2) must vanish, and hence (1) remains valid. The general case is now readily established. Given arbitrary zeros z I ,. . . ,zn inside or on the circle 121 = r, we form the function

Then F ( z ) is analytic and free of zeros in 1zI < r. Since IF(z)( = If(z)l whenever Izl = r, it follows from what has already been proved that log1 F(O)I

=

& jo2n

log(f(reiB)Ido.

But

and the result follows. I It is frequently useful to state Jensen's formula in a slightly altered form. If f(z) is analytic in 1zI < R, then we denote by n(r), 0 S r < R, the number of zeros z1 , z2,z3,.. . of f(z) for which Iz,I 5 r. Provided that f ( 0 ) # 0, it follows easily that

and Jensen's formula becomes

Jensen's formula provides a powerful tool for studying the relation

Pt. 1, Sec. 21

61

Jensen's Formula

between the growth of an entire function and the density of its zeros: the slower the growth, the more sparsely distributed the zeros. As an application of this principle, we shall apply Jensen's formula to the class of entire functions of exponential type.

Definition. An entire function f ( z ) is said to be of exponential type if the ineyuulity

holds for some positive constants A and B and all values of z.

Theorem 3.

If f ( z ) is an entire function of exponential type, then n(r)/r remains hounded as r -,GO. Proof. We may suppose without loss of generality that f(0) = 1. This is obvious if j'(0)# 0; if, on the other hand, f(z) has a zero of order rn at the origin, then we need only consider f(z)/z"'. Set N(r) =

j:

dt.

With this notation, Jensen's formula becomes

By assumption, I f ( z ) l 5 A21zl for some positive constants A and B and all values of z , so that

log(f(re")) 5 logA

+ Br

and hence also N ( r ) 5 logA

+ Br.

Since n(r) is a nondecreasing function of r, it follows that

f

n(r)log 2 = n(r)j,2r dt S

jr

2r

n(t) t dt 5 N(2r).

~

Entire Functions of Exponential Type

62

[Ch. 2

Thus, for all values of r, n(r) log 2 5 log A

+ 2Br,

and the result follows. I

Problems 1.

Using the calculus of residues show that

J: 2.

log sin 0 dO

= - 71 log

2.

Prove that if f ( z ) is entire and n(r) is the number of zeros of j ’ ( z ) in 5 r, then, even if f ( 0 ) = 0,

IzI

for some constant A that depends only on f ( z ) and not on r (r > 1). 3. Calculate n(r) for each of the entire functions ez - 1, sin z , and (sin &)I&. 4. Obtain an asymptotic estimate for n(r) when

3

Functions of Finite Order

To characterize the growth of an entire function f ( z ) , we introduce the “maximum modulus function”

Unless f(z) is a constant, M(r) is a strictly increasing function of r, and limr+mM(r)= 00. The first assertion follows from the maximum modulus principle, while the second is a consequence of Liouville’s theorem.

Pt. 1, Sec. 33

63

Functions of Finite Order

An entire function that grows no faster than a polynomial must in fact be a polynomial. More precisely, we have the following theorem.

Theorem 4. I f f ( z ) is un entire function for which M ( r ) 5 r"

fiir some integer n and all sufficiently large values of r, then f ( z ) is a polynomial qf degree at most n. Proof. Expand f ( z ) in a Taylor series,

and set p(z) = a,

+ a,z + . . . + a,z"

If

then g ( z ) is entire, and by hypothesis, g(z) 0 uniformly as IzI -+ a.It follows from the maximum modulus principle that g(z) must vanish identically, so that f ( z ) is a polynomial of degree no larger than n. I -+

Remark. I t is evident that the theorem remains valid (and the proof remains the same) under the weaker assumption that M ( r ) 5 r" on a sequence of circles IzI = rn with r, 00. -+

In light of Theorem 4, to measure the growth of a transcendental entire function, it is necessary to compare M ( r ) with functions that grow faster than every power of r.

Definition. An entire function f ( z ) is suid to be of finite order if r x i s t s a positive number k such that

there

M ( r ) 5 erk as soon us r is "sufficiently large", i.e., r > r(k). 7he greatest lower bound of

Entire Functions of Exponential Type

64

[Ch. 2

all positive numbers k for which this is true is called the order of the function und is denoted by p. Thus, p is the smallest nonnegative number such that

M(r) 5 elpts for every positive value of E , as soon as r is sufficiently large. It follows easily from the definition that the order of a nonconstant entire function is given by the formula p

=

lim sup r-m

log log M ( r ) log r

The order of a constant function is of course 0. Simple examples of functions of finite order include ez, sin z , and cos z , all of which are of order 1, and cos which is of order 3.An entire function of exponential type is of finite order at most 1; every polynomial is of order 0; the function e" is of infinite order. Theorem 3 is easily modified for functions of finite order (the details are left to the reader).

&,

Theorem 5. I f f ( z ) is an entire function of finite order p, then n(r) = O(rP+') for every positive number

E.

As a corollary, we have the following important result. Theorem 6. Iff ( z ) is an entire function of finite order p and i f zl, z 2 ,z 3 , . . . are its zeros, other than z = 0 , then the series

is convergent whenever

ci

> p.

Proof. We may suppose without loss of generality that the zeros of f ( z ) have been numbered so that

Pt. 1, Sec. 3]

65

Functions of Finite Order

Given a > p. choose fl so that p < fi c a . Since f ( z ) is of order p, it follows from Theorem 5 that n(r) 5 Ara for some constant A and all values of r. Take r = Iz,I ;then n(r) = n, and hence n 5 AlznlP The result follows at once since

for n = 1,2,3,. . . .

c:=,l/na’a <

00.

I

It is an important consequence of Theorem 6 that every entire function of finite order has a canonical factorization.

Corollary. I f f ( z ) is an entire function of finite order p , with zeros z l , z 2 ,z 3 , . . . , other than z = 0, then

where the product is a canonical product of genus p and p

5 p.

Proof. The remarks following Theorem 1 show that the product E(z/z,, p ) converges uniformly on each bounded region of the plane whenever

fl;=

and this is certainly true as long as p

+ 1 > p.

I

Much more is true. We shall prove subsequently (Theorem 9) that g(z) is a polynomial of degree no larger than p.

Definition. Let z l , z 2 , z 3 , . . . be a sequence of complex numbers, none of which is zero. The greatest lower bound of positive numbers a for which the series

66

Entire Functions of Exponential Type

[Ch. 2

is convergent is called the exponent of convergence of the sequence { zn} and is denoted by I . I f the series is divergent f o r every o! > 0, then we set I = m. For a “jinite” sequence, A = 0 by definition.

Example. The exponents of convergence of the sequences {en},

{nl”},

and

{logn}

are 0, I , and m, respectively. If I is the exponent of convergence of the sequence { z,,}, then the series o! > I and diverges if a < I ; if a = I , then no conclusion can be drawn. For example, the sequences { nilA ) and { ( n log’ n)”’} have the same exponent of convergence I , but l/lz,,(’ diverges for the first sequence and converges for the second. Suppose now that f ( z ) is entire of finite order p . Let I be the exponent of convergence of its zeros and let p be the genus of the corresponding canonical product. Our results thus far may be summarized by the following inequalities : by Theorem 6,

1 l/lznla converges if

1

I 5p

(Hadamard);

by the definitions alone,

If I is not an integer, then p = [I] (here [XI denotes the function “bracket x”,the largest integer not exceeding x); if I is an integer, then there is an ambiguity: p = I when the series when it converges.

1 1/12,,1’

diverges, while p

=

I

-

1

Problems 1. Prove that, for a given function, M ( r ) is a continuous function of r . 2. Determine M ( r ) for each of the entire functions ez, sin z, cos z, and (sin 3. Let f ( z ) be an entire function with f ( 0 ) = 1. Prove that

&I/&.

n(r) 5 log M(er),

where n(r) is the number of zeros of ,f(z) in the closed disk IzI 5 r

Pt. 1, Sec. 31

4.

Let

Functions of Finite Order

.f'(z)

67

be an entire function such that Re f ( z ) < M

5.

6.

7. 8. 9.

10.

for some constant M and all values of z. Show that f ( z ) is a constant. Determine the order of each of the entire functions eZZ,(sin &)/ and e''' d t . Show that the order of a product of two entire functions does not exceed the larger of the orders of the factors. Prove that f(z) and f ' ( z ) are of the same order. Determine the exponent of convergence of the zeros of eel - 1. Prove that if f ( z ) has at least one zero, but is not identically zero, then

&,

1;

If , f ( z )

=

a,z" is an entire function of order p, then

(the quotient is taken to be zero if a, = 0).(Hint: Suppose first that p < x and let the right-hand side of (1) be denoted by p. (a) P 5 P. Using Cauchy's estimate for the Taylor coefficients of f(z), show that for each fixed number k greater than p, one has (a,l

2 r er k

( n = 0,

-11

. .)

as soon as r is sufficiently large, i.e., r > r(k). Show that the quantity on the right achieves its minimum value when r = (n/k)'Ik, so that nik

( n large).

Conclude that k 2 p and hence that p 2 p. (b) I' 5 i'. If k > p, then

o j

n log n log(l/~a,~)

68

Entire Functions of Exponential Type

[Ch. 2

for all sufficiently large values of n, say n > N , so that lan\

s

(t)”i”

for n > N

Then

Write

where S , contains those terms with n < (2r)k and S , the remaining terms. Show that for all values of r, S , 2 1, while for all positive values of E ,

Conclude that p 5 k

+ E, and hence that p

p.)

11. Show that if the right-hand side of (1) in Problem 10 is finite, then f ( z ) is an entire function of order p. 12. Determine the order of each of the following entire functions:

00

(c)

C

e-n2zn,

n=n

(1‘

c r(i1an)

n=O

LY

>0

(Mittag-Leffler function).

13. Show that if E = 0, then the conclusion of Theorem 5 no longer holds.

Pt. 1, Sec. 41

4

69

Estimates for Canonical Products

Estimates for Canonical Products

Hadamard's inequality 1s p is valid for all entire functions. For canonical products, 1 and p are equal. Theorem 6 morel). 7he order of a canonical product is equal to the exponent of convergence of' its zeros. Proof. Let

be a canonical product of genus p and order p formed with the zeros zl ,z 2 , z 3 , . . . , and let I be the exponent of convergence of these zeros. Since 2 =< p . it is sufficient to show that p 5 1. In order to estimate the growth of P(z), we shall estimate the size of each of its factors. Fix z and write

where r

= IzI

and rn = IznI. For the second sum

c2we use the inequality

valid whenever IuI 5 $ (by virtue of (3) of Section 1). We then have

I f 1, = p

+ 1, then

If L < p hence

+ 1,

then I

+ c < p + 1 whenever

E

is sufficiently small, and

70

Entire Functions of Exponential Type

For the sum

I f ( u (2

[Ch. 2

X I ,suppose first that p > 0. Clearly,

f, then ( u (=<~ 2 p - k J ~ J p( k

=

1,. . . , p),

and hence log(E(u,p ) ( 5 log(1 - u (

+

2p(u(p

5 2 p + '(u(p,

whenever p > 0 and (uI 2

f. Therefore, for each E > 0,

for every positive number

E,

and ( 2 ) remains valid when p is replaced by c.

The proof is over: combining (1) and (2), we see that

for every c > 0, so that p 5

A. I

The simplest canonical product is of genus zero:

n ): ,rj

P(z) =

(1 -

n=

with

'cc

I

I

By Borel's theorem, P ( z ) is of order at most 1. However, a lot more can be said: P ( z ) is actually an entire function of exponential type. We first state the following definition.

Pt. 1, Sec. 41

71

Estimates for Canonical Products

Definition. Let f ( z ) be an entire function of exponential type. The exponential type? of f ( z ) is defined to be the number

The zero function has exponential type 0, by convention. According to the definition, k is the smallest nonnegative number such that

for any given E > 0 as soon as r is sufficiently large. The functions ez, sin z, cos z, and (sin z)/z are all o f exponential type 1. Any entire function of order p < 1 is of exponential type 0.

Theoiem 7. A canonical product of genus zero is an entire function of exponential type zero. Proof. Let P(z) be the canonical product m

with

"

1

-<

00.

n = l lznl

n=l

We may suppose that the zn are numbered in nondecreasing order of magnitude. In this case the sequence {l/lznl} is nonincreasing, and the convergence of the series C l/lznl implies that n -+O lzn

I

as n + m .

t The notion of "exponential type'' is not to be confused with that of "type". An entire function of positive order p is said to be of type T (relative to that order) if

Thus, for example, the entire function (sin it is clearly of exponential type 0 .

fi)/f i is of order p

=

4 and type 7 = 1, whereas

72

Entire Functions of Exponential Type

[Ch. 2

It follows readily from this that if n(r) denotes the number of points zl, z 2 , z3,.. . for which Iz,I 5 r, then

(see Problem 1). l / \ z n \ as a Stieltjes integral and then integrating by parts, Writing we obtain

1

Hence the integral n(t)

joF d t

(4)

is convergent. We can now estimate the growth of P ( z ) as follows: if r

=

I;

log( 1

r

t(t

= 121,

then

+ f) d n ( t )

+ r ) n ( t )dt

(integrate by parts)

From this estimate, together with ( 3 ) and the convergence of the integral (4), we readily obtain the asymptotic inequality

for each fixed c > 0, as soon as r is sufficiently large. This shows that P ( z ) is of exponential type zero. I

F't. 1, Sec. 41

73

Estimates for Canonical Products

The next theorem shows that the maximum modulus of l/P(z) is very often of the same order of magnitude as the maximum modulus of P ( z ) .

Theorem 8. I f P ( z ) is a canonical product of order p. then for each

on circles

131

=

E

> 0,

r of arbitrarily large radius.

Proof. Let P ( z ) be a canonical product of genus p and order p, formed with the zeros z , , z 2 , z 3 , .. . . Fix h, h > p, and let D, denote the disk ( z - z,(

2

Iz,I-h,

n

=

1,2,3,..

It is to be shown that (5) holds for each E > 0 whenever z lies outside every disk D, and r = IzI is sufficiently large, i.e., r > r ( E , h). Since the sum of the radii of these excluded disks is finite, the domain C - u Dn is nonempty and certainly contains circles IzI = r of arbitrarily large radius. Put r, = Iz,,~. The same method used to prove Theorem 6 shows that for every I-; > 0,

(remember that p then

Therefore, for each

1 rn 5 2r

A). If z lies outside every excluded disk and if r, S 2r,

=

log11 -

I:

f -n

> 0 and all sufficiently large values of r ,

1

2 -(h

+ l).log(2r).n(2r)

2 -(h

+ 1).10g(2r).ri'+~

(by Theorem 5)

Combining (6) and (7), we have the desired result. I

Entire Functions of Exponential Type

74

[Ch. 2

Problems 1.

2.

Prove: if {A,} is a nondecreasing sequence of positive numbers and if n(r) denotes the number of A,, not exceeding r, then the statements lim,,,, n/;Z,, = L and limr+mn(r)/r = L are equivalent. Determine the order of each of the canonical products and

i ( l

n= 1

n= 1

3. Prove that J ' ( z ) = k if and only if

+ :),

with a > 1.

u,z" is an entire function of exponential type

k = lim sup

< m.

n+ an

4. Prove: f ( z ) and f ' ( z ) are of the same exponential type. 5. Give an example of an entire function of order 1 that is not of expo-

nential type. 6. Let r l , r z , r 3 , .. . be a sequence of positive real numbers. Show that for l/r; and the integral 1 "; n(t)/ each positive number a, the series t '+ dt converge or diverge together.

I;:=,

'

5

Hadamard's Factorization Theorem

Equipped with the results of the preceding section, we can now establish the fundamental factorization theorem for entire functions of finite order. It is due to Hadamard who used the result in his celebrated proof of the Prime Number Theorem. It is one of the classical theorems in function theory. Theorem 9 (Hadamard). and if

If f ( z ) is an entire function of finite order p

f ( z ) = z"e~"'P(z) is its canonical factorization, then g(z) is a polynomial of degree no larger than p. Proof.

Let

A denote the exponent of convergence

of the zeros of the

Pt. 1, Sec. 51

75

Hadamard's Factorization Theorem

canonical product P ( z ) . Then P(z) is of order ,Iand ,I5 p . Let trary positive number. By assumption, Ioglf'(z)l < as soon as IzI

=

E

be an arbi-

rPtC

r is sufficiently large, while, by Theorem 8,

IoglP(z)l > -r'+'

>

-rP+'

on circles (zI = I' of arbitrarily large radius. Combining these two inequalities, we see that

< 2rP+' on a sequence of circles IzI = r, with r, + CLI. Since g(z) is entire, the BorelCaratheodory inequality (see Titchmarsh [1939, p. 1741) shows that

on the same sequence of circles ( z ( = r,. Conclusion: g(z) is a polynomial of degree at most p + I: (see the Remark following Theorem 4). Since E was arbitrary, the result follows. I

Remarks

1. p = max(A, deg y). Indeed, we now know that both A and the degree of g(z) d o not exceed p , so that max(l, deg g) 5 p. But the order of a product of two entire functions cannot exceed the larger of the orders of the factors (Problem 6, Section 3). Since zrnP(z)is of order 1 and eq(')is of order equal to the degree of g(z), we have p 5 max(1, deg g), and the result follows. 2. If p is not an integer, then A = p (for in this case the degree of g(z) is strictly smaller than p), and hence the form of the canonical product is uniquely determined: the genus p is equal to [p]. If, on the other hand, p is an integer, then there is an ambiguity: p may be equal to p or p - 1. For example, the canonical products

fi ( I n= I

-

f)8"

and

fi (1 -) -

n=2

are both of order 1, but of different genus.

Z

76

Entire Functions of Exponential Type

[Ch. 2

The following impressive corollary is a direct consequence of Hadamard’s theorem.

Corollary. An entire function of nonintegral order assumes every finite value infinitely many times. Proof. Since f ( z ) and f ( z ) - c have the same order for each constant c, it is enough to show that an entire function of nonintegral order has infinitely many zeros. But this is immediate from Remark 2 above since, in this case, the exponent of convergence must be positive. I

Example 1. Consider the entire function

It is of order p = and has a simple zero at z fore, by Hadamard’s theorem,

Since f (0) = 1, it follows that c

=

=

n2 (n = 1,2,3,. . .). There-

1. Observe that when z is replaced by

z 2 , we obtain the familiar formula

Example 2. The gamma function T(z) is defined by the formula

where y is Euler’s constant,

It is clear that T(z) is meromorphic in the entire plane and has simple poles at z = 0, - 1, -2,. . . . The constant y appearing in the exponent is

Pt. 1, Sec. 51

77

Hadamard's Factorization Theorem

chosen so that

r(1)= 1. To see this, simply observe that

Then P ( z - 1) is an entire function of order 1 and has (simple) zeros at z = 0, - 1, -2,. ... By Hadamard's theorem, we can write P(z - 1) = zeAZ+BP(z). The values of A and B are easily determined. Taking the logarithmic derivative of both sides, we obtain - - 1) = - +1 A + -.-l+n n z 1

f(L-k), n='

z+n

which reduces to A = 0. Setting z = 1, we find

so that eB = ey, and hence P(z

- 1) = zeYP(2).

It follows readily from this that T ( z ) satisfies the difference equation

Since

r(1) = 1,

repeated application of the formula above shows that

for every positive integer n. Thus, T ( z ) is an extension of the factorial function to nonintegral values of the argument.

Entire Functions of Exponential Type

78

[Ch.2

From the relation

P(z)P(- z )

sin n z

= -,

XZ

we derive the important identity

Problems Use Hadamard's theorem to determine the canonical factorization of cos 2. Show that 1.

4.

XZ

712

4

4

cos--sin-=(1

-z)

3. Let f ( z ) be an entire function of finite order p, with zeros z , , z 2 , z 3 , . . . other than z = 0. Let p be a nonnegative integer such that

n= 1

and write

(Note that the product is not necessarily a canonical product.) Does it still follow that g(z) is a polynomial? 4. Prove: if f ( z ) is an entire function of finite order p , but not identically zero, then for each E > 0,

on circles IzI = r of arbitrarily large radius. 5. Show that r(i)=

&.

Pt. 1, Sec. 51

Hadamard’s Factorization Theorem

79

6. Show that

+ i) +

=

1 . 3 . 5 . . . ( 2 n - 1) Ji 2”

( n = 1 , 2 , 3 ) . . .).

7. Show that

( H i n t : Put f(z)

=

r’(z)/I-(z). Show that

j’(z)

+ f’(z + +) = 2f’(2z),

and then integrate.) 8. Prove that “

1

9. Establish Legendre’s duplication formula,

-

10. Show that l/T(z) is not of exponential type. ( H i n t : Use Stirling’s formula to show that log M ( r ) r log r as r + 00.) I I . Prove Laguerre’s theorem: if f(z) is a nonconstant entire function, real for real z, of order less than 2, and with real zeros only, then (1) the zeros of f’(z) are also real and (2) f’(z) vanishes once and only once between successive zeros of f(z). ( H i n t : For the first assertion, show that

Im(

E)

=0

For the second, show that

only when Im z

= 0.

80

Entire Functions of Exponential Type

PART TWO.

[Ch. 2

RESTRICTIONS ALONG A LINE

For the remainder of this chapter we shall be concerned mainly with the class of entire functions of exponential type. Our primary aim is to characterize those functions of this class that belong to L2 along the real axis. These functions play an important role in both the theory and applications of entire functions. If an entire function of exponential type is known to satisfy additional growth conditions along a line, then more can be asserted about its growth in general and hence about the distribution of its zeros. We shall show, for example, that boundedness on one line carries with it boundedness on every parallel line; a similar assertion is true for functions belonging to L p on a given line. These results are in marked contrast with the behavior of entire functions in general, about which such assertions are not universally true. We begin by establishing a far-reaching generalization of the maximum modulus principle.

1 The "Phragmh-Lindelof"

Method

Theorems that come under this collective heading establish the boundedness of an analytic function inside an infinite region, given that the function is known to be bounded on the boundary and not of too rapid growth inside. The fundamental result is that in which the unbounded region is an infinite sector.

Theorem 10 (Phragmbn-Lindelof). Let f ( z ) be continuous on a closed sector of opening nfa and analytic in the open sector. Suppose that on the bounding rays of the sector,

and that for some

fl < a,

whenever z lies inside the sector and IzI = r is sufficiently large. Then (z)l S M throughout the sector.

If

PrmJ

We can suppose without loss of generality that the given sector

Pt. 2, Sec. 11

The ”Phragmen-Lindelof‘ Method

81

is symmetric with respect to the positive real axis and has its vertex at the origin. We introduce the auxiliary function

where /I < y < a and E > 0. Here z7 denotes that single-valued analytic branch of the multiple-valued function zy = exp(y log z ) that takes positive values for positive real z . Setting z = re’”, we have

On the bounding rays of the sector, cos y0 is positive, and hence

On the arc 101 5 n/2a of the circle ( z (= r ,

as soon as r is sufficiently large. But this last expression approaches zero as r + a,so that, if r is sufficiently large, Ig(z)l S M on this arc also. By the maximum modulus principle, Ig(z)l M throughout that part of the sector for which Izl 5 r and hence throughout the entire sector since r can be made arbitrarily large. Therefore, we conclude that

everywhere within the sector. Since E was arbitrary, the result follows. I The theorem is sharp in the sense that the conclusion is no longer valid when /j = a. Indeed, we need only consider the function

f(z)

is bounded on the rays of the sector but certainly not in the interior.

Corollary. An entire function of’ order less than one that is bounded on u line must reduce to a constant.

We can now prove that an entire function of exponential type that is

82

Entire Functions of Exponential Type

[Ch. 2

bounded on a line must be bounded on every parallel line. There is no loss of generality in supposing that the given line is the real axis.

Theorem 11. Let f ' ( z ) he an entire function such that

f o r all real values of x . Then

Proof. Suppose first that y > 0. Let c be an arbitrary positive number and put y(z) =

ei(B+E)Zf'(z).

for all real x, while y(iy)+O

as y + m.

Let N denote the maximum value of (y(z)(on the nonnegative imaginary axis. The Phragmen-Lindelof theorem, applied separately to the first and second quadrants, shows that

throughout the upper half-plane. A simple application of the maximum modulus principle then shows that N S M, and hence lg(z)(

5M

whenever

I m z > 0.

Pt. 2, Sec. I]

The "Phragmh-Lindelof" Method

83

throughout the upper half-plane, and the result follows by letting E --t 0. The case in which y < 0 can be reduced to the first case by considering ./'( - z ) . I We note that the theorem is sharp in the sense that neither B nor M can be replaced by any smaller constant in the conclusion-simply consider ,/'(=) = sin:. We note also that the conclusion holds under the apparently weaker assumption that f ' ( z ) is an entire function of exponential type at most B. For in this case we have

for each positive number

t:

and all values of z , and consequently

Now let I: + 0. Theorem I I shows that an entire function f ( z ) of exponential type that is bounded on the real axis is in fact "uniformly bounded" in every horizontal strip. If, in addition, we know that .f'(x)-tO as 1x1+ x.,then more can be asserted.

Theorem 12. !/' j ' ( z )is un entire jiinction uf exponentid type and iJ'

i~tz(/orti~Iy in elwry horizontal strip.

Proof: Since j ' ( z ) is uniformly bounded in every horizontal strip, the result is an immediate consequence of Montel's theorem (see Titchmarsh [1939. p. 1701). I

The Fourier St ieltjes integrals

S'

eiZ1dtfJ(t ), T

a4

Entire Functions of Exponential Type

[Ch. 2

where w(t) is a function of bounded variation on the interval [ - T , T ] , provide a large class of examples of entire functions of exponential type (no larger than T) that are bounded on the real axis. Special cases include the "almost periodic" exponential sums

n

with

--t

I , 5 T and cIc,I <

as well as the integrals

00,

s'.

e"'.f'(t)d t ,

with S Y , If(t)l dt < co. The totality B, of all entire functions f(z) of exponential type at most that are bounded on the real axis, together with the norm

t

constitutes one of the important classical Banach spaces of entire functions. That B, is a vector space (under pointwise addition and scalar multiplication) is clear; that it is a Banach space is an easy consequence of Theorem 11. The space B, was first considered by Bernstein. The following remarkable theorem is due to him: if f ( z )E B,, then

moreover, equality can hold only for functions of the form

where a and p are constants. Bernstein's inequality, as the aforementioned result is known, plays an important role in the theory of approximation of continuous functions (see Akhiezer [1956]; see also Problems 12 and 13 and Example 2 of Section 4 for a proof of Bernstein's inequality).

Problems 1.

Let f ( z ) be continuous in the closed square [ - 1, I] x [ - 1, 11 and analytic in the interior. Suppose that on each bounding edge Si

Pt. 2, Sec. 11

The "Phragmbn-Lindelof" Method

85

( i = I, 2,3,4) we have IJ'(z)l

2.

5 Mi

whenever z E Si.

Prove that Ij'(0)l 5 ( M l M z M 3 M 4 ) 1 / 4 . Prove that the conclusion of Theorem 10 still holds if we are only given f ( z ) = O(ecr'")

3. 4. 5.

6.

for each c > 0, uniformly in the sector. (Hint: Consider the function e - f , z a j( z ) . ) Prove that an entire function of exponential type zero that is bounded on a line must reduce to a constant. Prove that an entire function of order p < that is bounded on a ray must reduce to a constant. Prove that an entire function of exponential type that is bounded on two nonparallel lines must reduce to a constant. Let / ' ( z )be continuous in a closed sector and analytic and bounded in its interior. Prove that if f ( z ) + u as z+m along the bounding rays of the sector, then f ( z ) + a uniformly in the sector. (Hint: Suppose first that larg z ( 5 ct < 4 2 and consider the function ( z / ( z A ) ) f ( z ) , with 1. > 0.) Let f ( z ) be continuous in a closed sector and analytic and bounded in its interior. Suppose that j ' ( z ) + u as z - 0 0 along one of the bounding rays, while , f ( z ) + h as z cx. along the other. Prove that u = h. ( H i n t : Consider the function ( f ( z ) - $(a b))'.) Let j ( z ) be analytic in an open connected set 0.Suppose that for each boundary point of R and each E > 0 there exists a neighborhood such that at every point z of R in this neighborhood one has

4

+

7.

--+

8.

(j(z)l< M

+

+

E.

Prove that I f ' ( z ) l 5 M throughout R (in fact, unless f ( z ) is a constant, 1.f ( z ) ( < MI. 9. Let I(:) be analytic and bounded in the right half-plane R e z > 0. Prove that if z 1 . ~ z , z 3 , ... are the zeros of j ( z ) in this half-plane, then the series

86

Entire Functions of Exponential Type

[Ch. 2

is convergent. ( H i n t : Prove that if I f ( z ) l 6 M whenever Rez > 0, then the stronger inequality

10.

holds for Re z > 0.) Prove that if f ( z ) is an entire function of exponential type, bounded on the real axis, and if z , , z 2 , z j r . , . are its zeros other than z = 0, then

( H i n t : Use the result of Problem 9.) Use Cauchy's integral formula to show that if f ( z ) is an entire function of exponential type that is bounded on the real axis, then f ' ( z ) is also bounded on the real axis. 12. Prove Bernstein's inequality for functions of the form

11.

where o(t)is of bounded variation on the interval [ - n,n]. ( H i n r : Use the relations

and

to show that

13. (Bernstein) If f ( t ) =

-N

cnein' is a trigonometric polynomial of

Pt. 2, Sec. 2]

87

Carleman‘s Formula

degree N and if M is the maximum value of l f ( r ) l (t real), then l.f”(r)l 5 M N for all real values off

2

Carleman’s Formula

Jensen’s formula provides a simple relation between the modulus of an analytic function on a circle and the distribution of its zeros inside the circle. If additional information is available on the growth of the function along a line, then Carleman’s formula is of fundamental importance. Theorem 13 (Carleman). Let f ( z ) be analytic for Im z 2 0 and let zk = rkeiHk ( k = 1, . . . n) he i t s zeros in rhe region ~

R = { z : l m z 2 0 und 1 S

(zI

5 R}

Then

where A ( R ) is a hourided function of R .

Proof. Suppose to begin with that f ( z ) has no zeros lying on dQ (the boundary of Q). We consider the contour integral

where dQ is assumed to be posirioely oriented and the integration begins at z = 1 with a fixed determination of the logarithm.

88

Entire Functions of Exponential Type

[Ch. 2

The method of proof is to evaluate I in two different ways and then equate the results. First

As we make one complete passage over the contour 82,log f ( z ) increases

by 2nin (recall that n is the number of zeros of f ( z ) inside first integral on the right-hand side is equal to

an),so that

The second integral is easily evaluated by the residue theorem-this

the

gives

Next, we evaluate I by integrating separately over each component of

an. On the positive real axis, we obtain

and on the negative real axis,

on the large semicircle, z = Re", and we obtain i

71R

:j

log(f(Re")) sin 0 d o ;

and, finally, on the small semicircle, we obtain a bounded function of R . The result now follows by equating the imaginary parts of both values of I . The restriction that f ( z ) be free of zeros on the contour can be eliminated by a simple continuity argument. The details are left to the reader. I An entire function f(z) of exponential type T > 0 that is bounded on the real axis is in many ways similar to sin TZ. It is clear, for example, that

Pt. 2, Sec. 2]

89

Carleman's Formula

f ( z ) must have an infinite number of zeros. Reason: By Hadamard's fac-

torization theorem,

if the product were finite, then ,f(z) could not remain bounded on the real axis without reducing to a constant. It is a simple consequence of Carleman's formula that the zeros of j ' ( z ) must cluster about the real axis. Specifically, we have the following result (cf. Problem 10, Section 1). Theorem 14. If' f(z) is an entire function of exponential type, bounded on the real uxis, arld if zn = rneion(n = I , 2 , 3 , . . .) are the zeros of f ( z ) other t h m z = 0. then the series

1 -.!!sin H n=1

IS

'n

absolutely convergeni

Proof. Since If(x)l is bounded on the real axis, the integrals

have a finite upper bound for all values of R > 1. In addition, since f ( z ) is of exponential type, there is a constant K such that loglf(ReiH)I5 K R , as soon as R is sufficiently large, so that

ii6:

2K loglf(Re'")l sin HdH 5 -. 7c

Applying Carleman's formula to f ( z ) in the upper and lower half-planes and then adding the result, we conclude that

Entire Functions of Exponential Type

90

[Ch. 2

for some constant A and all large values of R . But this implies that

rn < R / 2

and the result follows by letting R + co. I Let us write the conclusion of Theorem 14 as follows:

Then, for each positive number c , the zeros znk of f ( z ) that lie outside the sectors larg zI < E and larg z - ?II < E must satisfy “

1

Now, it can be shown that if f(z) is of exponential type t > 0 and bounded on the real axis, then the set consisting of all its zeros has a “density” equal to 2t/n, i.e.,

(this is hard; see, for example, Levinson [1940, Chap. 1111). Accordingly,

and hence “most” of the zeros of f ( z ) lie “near” the real axis. Theorem 14 may be reformulated as a uniqueness theorem: an entire function of exponential type that is bounded on the real axis is completely determined by its values on any set for which C l I m l/znl = 00. As an application, we shall prove the following completeness theorem for systems of complex exponentials in C[a, b ] .

Theorem 15. If {A,} is a sequence of distinct complex numbers jbr which larg A,, - n/21 I L < 4 2 and if

Pt. 2, Sec. 2]

91

Carleman’s Formula

then the system {e”.88‘\is eomplere in C[u, b]

fir

every finite interuul [a. b ] .

Proof: We argue by contradiction. If the system failed to be complete in C[u,b] for some interval [ u , h ] , then by virtue of the Riesz representation theorem we could find an entire function f ( z ) of the form

with o ( t ) of bounded variation on [ [ J , b ] , such that f ( z ) vanishes at every All but does not vanish identically. According to Theorem 14, the series

is absolutely convergent. But

and the contradiction proves the theorem. I By choosing the ,Itt to be purely imaginary, we obtain at once the following version of Muntz’s theorem on polynomial approximation. Corollary (Miintz). I f r e d numbers such that

(A,,A 2 , Jb3,. . .} is un increasing sequence

of positive

then the set q j powers II

t ,p

11

is complete in C[u, h] whenever 0 <

[A3

2 5

11

,‘“j

I

< b.

Problems 1.

Establish the following version of Carleman’s theorem: if f ( z ) is an-

92

Entire Functions of Exponential Type

[Ch. 2

alytic for I m z 2 0, if z l , . . ., zn are its zeros in the region {z:IzI < R, I m z > 0}, and if f ( 0 ) = 1, then

2. Show that Theorem 13 remains valid if f ( z ) is analytic for I m z > 0 and continuous for I m z 2 0. 3. Let f ( z ) be an entire function of exponential type and suppose that

for some constant M and all values of R > 1 . Prove that if zl, z2, z 3 , . . . are the zeros of f(z) other than z = 0, then

( H i n t : Set

4.

and integrate by parts.) Let f(z) be analytic for Rez 2 0 and suppose that for some positive number a,

uniformly in the sector. Show that f(z) is identically zero. (Hint: Consider the function g(z) = f(z) sin bz, where 0 < b < a.) 5. Let j ( z ) = [e"@(t)dt,

Pt. 2, Sec. 31

Integrability on a Line

93

where & t ) isa real integrable function on [a,h]. Show that if z1,z2,z3, ... are the zeros of f ( z ) other than z = 0, then

( H i n t : Show that e"'f ( z ) is bounded for Re z 5 0 and e-"'f'(z) is bounded for Re z 2 0.) 6. (a) Show that Miintz's theorem need not hold if a < 0. (b) Prove that the set { 1, l A 1 t, A 2 , ,. .} is complete in C[O, 13 whenever is an increasing sequence of positive numbers for which l/& = 'XI. ( H i n t : First show that the set { t ' l , t A 2 t"', , . . .> is complete in L2[0, 13 by considering the function

1

where 4 E L2[0,11 and Re z 2 0. Complete the proof by observing that for 0 5 t 5 1 and n = 1 , 2 , 3 , . . .,

3

Integrability on a Line

In this section we shall prove that an entire function of exponential type that belongs to Lp along the real axis must also belong to Lp along every line parallel to the real axis. In fact, much more is true. Theorem 16 (Plancherel-Polya). If f ( z ) is un entire function of exponentiul type T und iJ' fbr some positive number p ,

Entire Functions of Exponential Type

94

[Ch. 2

then

Plancherel and Polya [1938] have given an “elementary” proof of Theorem 16 based on the Phragmen-Lindelof method, and it is this proof that we reproduce here. The proof will require two preliminary lemmas. Let g(z) be continuous in the closed upper half-plane, Im z 2 0, analytic in the interior, and suppose that g(z) does not reduce to a constant. Let a and p be positive real numbers and put c ( z )=

1:.

lg(z

+ t)lPdt.

It is clear that G ( z ) is defined and continuous for I m z 2 0. Since lg(z)Ip is subharmonic for Im z > 0 (see Rudin [1966, p. 329]), so is C(z).

Lemma 1. Let g(z) be of exponential type in the half-plane I m z 2 0 und suppose that the following quantities are both finite:

M =

sup

G(x)

and

N = supG(iy).

- ma < x < m

Y’O

Then throughout this haljlplane, G(z)

5 max(M, 1).

Proof. Since g(z) is of exponential type, there exist positive numbers A and B such that Ig(z)l

For each positive number

5 AeBlzl E

for Im z 2 0.

(1)

define the auxiliary function

where A = The exponent of e appearing in (2) has two possible determinations in the half-plane Imz > 0; we choose the one whose real part is negative in the quarter-plane x > - a , y 2 0. Put

Pt. 2, Sec. 3]

95

Integrability on a Line

which is then defined and continuous in the upper half-plane Im z 2 0, and subharmonic in the interior. A simple calculation involving (1) and (2) shows that in the quarter plane x > - a , y 2 0,

where y = cos 3 ~ 1 8and ,

G,(z) 5 G(z)

for x 2 0, y 2 0,

and in particular G,(x)

S M for

x

20

G,(iy) 5 N

and

for y 2 0.

Let zo be a fixed but arbitrary point in the first quadrant. We shall apply the maximum principle to GJz) in the region fi = {z:Rez 2 0, Im z 2 0, IzI 5 R } , choosing R large enough so that (i) z,, E R, and (ii) the maximum value of G,(z) on fi is not attained on the circular arc(z1 = R (this is possible by virtue of (3)). Since G,(z) does not reduce to a constant, the maximum value of G,(z) on R must be attained on one of the coordinate axes, and hence, in particular, G,(z,)

5 max(M, N ) .

Now let I: -, 0. This establishes the result for the first quadrant; the proof for the second quadrant is the same. I Lemma 2. I n urldition to the hypotheses of’ Lemma 1, suppose that lim y(x

)”*

unifbrmly in x, ,fir

--N

+ iy) = 0

(4)

5 x 5 a . Then

G(z) 5 M

whenever

Proof. It is sufficient to show that N

Im z 2 0.

5 M . By virtue of (4),we see that

96

Entire Functions of Exponential Type

[Ch. 2

the function G(iy) approaches zero as y + co, and so must attain its least upper bound N for some finite value of y, say y = y,. If y , = 0, then

N = G(iy,) = G(0) 5 M If, on the other hand, y, > 0, then the maximum principle shows that the least upper bound of G(z) in the half-plane I m z 2 0 cannot be attained at the interior point z = iy,. Therefore, by Lemma 1,

N

= G(iy,)

< max(M, N ) ,

and so N < M. I Theorem 16 can now be established. Proof of Theorem 26. It is sufficient to prove the theorem when y > 0 and f ( z ) is not identically zero. Let E be a fixed positive number and consider the function g(z) = f(z)e"'+'''. It is a simple matter to verify that for each positive number a, the functions g(z) and G(z) fulfill all of the necessary hypotheses of Lemmas 1 and 2. Consequently, if y > 0, then by Lemma 2

This together with the definitions of g(z) and G(z) implies

The proof is over: first let a + co,then let

E

--t

0. I

An important corollary to Theorem 16 asserts that f ( x )+ 0 as 1x1 ccj whenever f ( z ) is of exponential type and belongs to Lp along the real axis. The next theorem will prove even more. +

Theorem 17. Let f ' ( z ) be an entire function

of

exponential type T and sup-

Pt. 2, Sec. 31

Integrability on a Line

97

pose thut j b r some positive number p ,

lj’ (A1,; is

[in

increusiny sequence uf r e d n u m h m such thut

then

where B is a constant that

Proof: Since

I.f’I”

depends

onlj1

on p ,

T, und E .

is subharmonic, we know that the inequality

holds for all values of r (see Rudin [1966, p. 329]), and hence that

for every zn and every 6 > 0 (multiply both sides of (5) by r and integrate between 0 and 6). Therefore,

1

r d

Pd

98

Entire Functions of Exponential Type

[Ch. 2

+

Take 6 = 4 2 . Then the intervals (,In - 6,,lf1 6) are pairwise disjoint, and the last expression above is no larger than

Applying Theorem 16, we conclude that

where B = B(p, r, 6). I Remark. A sequence {%,,} of real or complex numbers is said to be separuted if for some positive number E ,

[Al,

-

%,I,

2

E

whenever

n#m

The theorem is easily modified to allow for complex 3,,,’s, provided they are separated and

The details are left to the reader

Corollary. If f ( z ) is an entire function of exponential type and if

Problems 1. Give an example of an entire function that belongs to L’ along the

real axis but does not belong to L’ along any other line.

Pt. 2, Sec. 3]

99

Integrability on a Line

2.

Prove that if j ( z ) is an entire function of exponential type zero and J1, I f ’ ( x ) l ” d.u < x for some positive value of p , then f ( z ) is identically zero. 3. Verify the Remurk following Theorem 17. 4. Prove: if f ( 2 ) is an entire function of exponential type T and

for some positive value of p , then

where C is a constant depending only on p and 6 >o,

T , (Hint:

For every

5 . Show that the space E; ( p 2 1, T > 0) consisting of all entire functions f ( $ of exponential type at most T for which

is a Banach space. 6 . Show that if & t ) E L 2 [ - - T 5 ,

7.

TI, then

belongs to Ef (see Problem 5 ) . (Plancherel-Polya) If f ( z ) is an entire function of exponential type T and j’ I f ( x ) I P d x< for some positive value of p , then j ’ ( z ) has the same properties. Moreover, j?AI f ’ ( x ) l p d x 5 A f Z W I j ( x ) l p d x , where A depends only on p and T. (Hint: If 6 > 0, then

100

Entire Functions of Exponential Type

[Ch. 2

where C depends only on p and 6. To prove this, apply inequality ( 5 ) to the function

inside a circle of radius r centered at z

=0

and obtain

Now multiply both sides by rP+' and integrate with respect to Y between 0 and 6. (With Theorem 18 of the next section, the problem becomes trivial for p = 2; see the remarks preceding Theorem 19.))

4

The Paley-Wiener Theorem

It is a relatively simple matter to exhibit a large class of examples of entire functions f ( z ) of exponential type for which

J -m

If 4 E L 2 [ - A, A ] , then

jA

f(z) =

A

4(t)eiz'dt

is such a function : a straightforward application of Morera's theorem shows that f ( z ) is entire; the estimate

shows that . f ( z ) is of exponential type at most A ; and Plancherel's theorem l f ( x ) l 2 d x = 2nIA, l4(t)I2dt < 00. shows that It is a remarkable fact that there are no other examples-every entire function of exponential type that belongs to L2 along the real axis is obtained in this way. This is the content of the celebrated "Paley-Wiener Theorem".

I?m

Pt. 2, Sec. 41

The Paley-Wiener Theorem

Theorem 18 (Paley-Wiener).

101

Let f ( z ) be an entire function such that

for positive constants A and C and all values of' z , and

f:, Then there exists a junction

( f ( x ) I 2 d x< m.

4 in

L 2 [ - A, A ] such that

Proof (Boas). Let &t) be the Fourier transform of f ( x ) , i.e.,

where the integral is to be interpreted as a limit in the mean in the L2 sense. Then 4 E L2(- x,K ) and, by the Fourier inversion formula, f ( x ) = J;,x

&t)e'"' d t .

Accordingly, we need only show that 4(t ) vanishes almost everywhere outside the interval [ - A , A ] . This will establish (2) for all real values of 2 and hence for all complex values as well since both sides of (2) are entire. Let T be a positive number. We shall consider the contour integral 1=

s

.f(z)e-'Z'dz,

where t is tixed and y consists of the upper three sides of the rectangle [ - T , TI x [0, T I . Since the integrand is an entire function of z for each value o f t ,we have I = -

f (x)e

jX'

dx

by Cauchy's integral theorem. We are going to show that 1 + O as T

--+

%8

Entire Functions of Exponential Type

102

[Ch. 2

whenever It1 > A . Since

it will follow that &t) = 0 almost everywhere outside [ - A , A ] . Suppose first that t < - A . Integrating over each of the three sides of y , we readily obtain

=I,

+ + I,. 12

We shall estimate the size of each of these three integrals. Since j?m, I.f(x)12 dx is finite, it follows that f ( x ) -0 as 1x1 + 00 (by the corollary to Theorem 17), and hence, in particular, that I f ( x ) ( has a finite upper bound M on the real axis. By Theorem 11, If(x

+ iT)I 5 MeAT

for all real values of x, and hence l 2 5 2TMe"+A'T.

Since t < - A , the exponent above is negative, and we conclude that I2+Oas T+m. Turning next to I , ( I 3 is treated in the same way), we write

Theorem 12 shows that for each fixed R ,

unifbrmly in y , 0 S y 5 R . Consequently,

Pt. 2, Sec. 41

103

The Paley-Wiener Theorem

+

It remains only to show that jg e''I.f(T iy)l dy can be made arbitrarily small by choosing R and T sufficiently large. Again appealing to Theorem 11. we have

and since t + A is negative, the last term approaches zero as R and T approach infinity. We have thus shown that I + O as T + 'XI whenever t < - A . The case t > A is treated similarly, with y in the lower half-plane. The details, which are obvious, are left to the reader. I It is important to note that condition (1) may be replaced by the seemingly weaker assumption that f ( z ) is entire of exponential type at most A . For, if this is the case, then

for every positive number c, and the proof of Theorem 18 shows that the Fourier transform of f(x) must vanish almost everywhere outside the interval [ - A - c, A E ] , and so almost everywhere outside [ - A , A ] . The following useful corollary is now readily established.

+

Corollary. If' j" ( z ) is an entire junction of' exponential type A , belonging to L2 on the real axis. then

The Paley-Wiener theorem has a wide range of important and varied applications, several of which are illustrated by the following examples. We begin with an integral representation for entire functions of exponential type that are bounded along the real axis. Example 1. If j ( z )E B,, then f(z) = f(0)

where

4 E L z [ --5, TI.

+z

j;,

&t)e'Z' d t ,

104

[Ch. 2

Entire Functions of Exponential Type

Proof. It is clear that the function

is entire of exponential type at most z and belongs to L2 along the real axis. Now apply Theorem 18. I Example 2 (Bernstein’s inequality). If f ( z )E B, and If(x)I real values of x, then (f’(X)(

sM

for all

5 .rM.

Proof. The inequality is easily established for functions of the form

SI

eizfdw(t),

(3)

r

where w(t) is of bounded variation on the interval [ -z, T] (see Problem 12, Section 1). Suppose now that f ( z ) is an arbitrary function belonging to B,. If we put

+

then g,(z) is entire of exponential type at most 7 E and belongs to L2 along the real axis. By virtue of the Paley-Wiener theorem, g,(z) is of the form (3) with T + E in place of z. Since Ig(x)l 5 M , it follows that

for every real x. But g : ( x ) - + f ‘ ( x ) as E + O , and the result follows. I As a final application, we shall establish an interesting quadrature formula for a certain class of entire functions of exponential type.

Example 3. If f ( z ) is an entire function of exponential type at most 2n and !Ern If(x)l d x < co, then a) n=-m

Proof. Since f ( x ) is integrable over ( - cu,a), so is f ’ ( x ) (Problem 7,

Pt. 2, Sec. 51

105

The Paley-Wiener Space

Section 3) and f (x) + 0 as 1x1 + x. Under these conditions, the Poisson summution ,formula is applicable: if F is the Fourier transform of ,f, i.e., if Ffr) = (1/3n){" II f ( x ) e - ' " ' d x , then

c

c

f ( n ) = 27c

F(27cn).

But S I I f ( x ) ( ' dx < co, and the Paley-Wiener theorem implies that F ( t ) = 0 for rurry t with It1 2 2n (note that F ( t ) is continuous). Conclusion: JL

f ' ( n ) = 2nF(O) =

Im

f(x)dx. I

Problems Let j ' ( z ) = J A A 4(r)riz'dt,where ~ E L ' [ - A A , ] , and suppose that 4 does not vanish almost everywhere in any neighborhood of A . Show that j ' ( z ) is of order 1 and exponential type A . 2. Establish the corollary to Theorem 18. 3. Prove the following "gap" theorem for entire functions: if f ( z ) is an entire function of exponential type, bounded on the real axis, and if 1.

.f '"'(0)

=0

for a sequence of positive integers An for which

1

1

-=m

and

1 h1 = m , Inodd

1. even A n

n

then f(z) must reduce to a constant. ( H i n t : Put g(z) = (f(z) - f(O))/z. Then g(z) = SAA 4(()eizf d t , with 4 in L'[- A, A ] . Consider the functions JoA

5

+ 4( - t ) )dt

tz(4(t)

and

The Paley-Wiener Space

The totality of all entire functions of exponential type at most 7c that are square integrable on the real axis will henceforth be known as the

106

Entire Functions of Exponential Type

[Ch. 2

Paley-Wiener space and will be designated by P. Clearly, P is a vector space under pointwise addition and scalar multiplication; it is also an inner product space with respect to the inner product

Since the Fourier transform is an isometry, the Paley- Wiener theorem shows that P is a separable Hilbert space, isometrically isomorphic to L2[ - n,n ] . The isomorphism between P and L 2 [- n, n] has far-reaching consequences. At present, we shall investigate some of the ways in which known properties of L 2 [-n, n] can be transformed easily into nontrivial assertions about P. This will then serve as the basis for a more penetrating analysis of nonharmonic Fourier series in L 2 [-n, n]. If f belongs to P and has the representation f(z)

with

I$

=

2n

s'

-n

4(t)ei" d t ,

in L 2 [-n, n ] , then Plancherel's theorem shows that

When Parseval's identity is applied to

4, we obtain

the important formula

By taking the Fourier transform of ein' ( n = 0, 1, f 2 , . . .), we see that the set of functions {(sinx(z - n))/n(z - n)}mm forms an orthonormal basis for P. Accordingly, each function f in P has a unique expansion of the form a.

c

f ( z ) = n = - , n c,

sin n(z - n) n(z - n) '

with Clcnl2 < co. The convergence of the series is understood to be in the metric of P. But convergence in P implies uniform convergence in each

Pt. 2, Sec. 51

107

The Paley-Wiener Space

horizontal strip. This is an immediate consequence of the following useful estimate :

I f ( x + iy)l s enll’l1 f 11.

(2)

(Proof: Simply use (1) together with the fact that ) 1 $11 = I l f l l . ) If we now set z = 11, it follows that c, = f ( n ) , and we have thereby obtained the cardinal series for f :

Thus, a function in P can be recaptured from its values at the integers. The cardinal series was first introduced by Whittaker [1915]. It plays an important role in the mathematical theory of communication, where it is known as the sampling theorem (see Shannon [1949] and Shannon and Weaver [1964]). The Paley- Wiener representation yields a trivial proof that P is “closed” under differentiation (cf. Problem 7, Section 3). Indeed, if f ( z ) is given by (I),then

and t&t) E L 2 [ - x , n]. Moreover, we have the following sharp estimate for the norm of .f’(z) :

Theorem 19. The Puley-Wiener space is a functional Hilbert space of entire ,functions. I t s reproducing kernel K is given b y K(z,W ) =

sin x(z - W) x ( z - W) ’

and the integral representation

sin x(t - z ) dt x(t - z ) is valid ji)r every function f belonging to the space.

108

Entire Functions of Exponential Type

[Ch. 2

Pruuf. Inequality (2) shows that “point-evaluations” on P are bounded linear functionals. Thus, P is a functional Hilbert space. If f(z) E P,then the Paley-Wiener theorem shows that

for some function 4 in L2[-n,n] and all values of w . But the Fourier transform is an isomorphism between L 2 [ - n , n] and P, and hence sin n(z - W) n(z - W) Since K(z, w) is the unique function on C x C for which f(w)

=

(f(Z), K(z, w)),

it follows that K(z, w) = (sin n(z - W))/n(z- W). I

Corollary. Ij f € P and z = x

+ i y , then

Problems 1. Show that if f ’ E P,then

2.

Let f(z) be an entire function of exponential type at most n, and suppose that f ( x ) = O(xU),

Show that

0 < c( <

4..

Pt. 2, Sec. 51

The Paley-Wiener Space

3. Obtain the cardinal series for functions of the form f'(z)=

2rc

f 4(t)e-'"

dt,

~n

with 4 integrable on [-n,n]. (Hint: Replace over [ -71, rc] and integrate term-by-term.) 4. Show that the function

l'M = "c =I

4 by its Fourier series

( - 1)" sin n(z - n) ~

n

n(z - n)

belongs to the Paley-Wiener space but that zf(z) is not bounded on the real axis (cf. Example I , Section 4). 5. Establish the corollary to Theorem 19.

This Page Intentionally Left Blank

3

THE COMPLETENESS OF SETS OF COMPLEX EXPONENTIALS

In this chapter we shall discuss a few of the fundamental completeness properties of sets of complex exponentials {eiAmr}over a finite interval of the real axis. The most extensive results in this direction were obtained by Paley and Wiener [1934] and Levinson [1940]. At the same time, we will be laying the groundwork for a more penetrating investigation of nonharmonic Fourier series in L 2 . By making an appropriate change of variables, we may consider the system (elAn')over an interval [-A, A] symmetric with respect to the origin. The spaces of interest will be L P [ - A , A], for 1 5 p < m, and C[ - A, A]. As we have seen, for the system {eiAn'} to be incomplete in C [ -A, A], it is necessary and sufficient that there exist a nontrivial entire function f ( z ) , zero for every in, and expressible in the form A

f(z) =

J-A

eizrh ( t ) ,

where ~ ( tis) of bounded variation on [ - A , A]. When Lp[- A, A] is considered, , f ( z ) takes the form

+

with 4 in L"[ - A, A]. Here q denotes the conjugate exponent: l/p l/q = 1 . The remarks above establish a connection between completeness and uniqueness. If there are enough 1"'s to guarantee that f ( z ) vanishes identically whenever ,f(j.,,) =

0

(n 111

=

1, 2, 3 , . . .),

112

The Completeness of Complex Exponentials

[Ch. 3

i.e., if (A,} is a set of uniqueness for f(z), then the system of exponentials { eiAnr}will be complete in the corresponding space. Therefore, every theorem on uniqueness generates a theorem on completeness, and conversely. In L 2 [ - n n , n ] ,the situation is particularly simple. Here the system of exponentials {eiLpi'}is complete whenever {A,} is a set of uniqueness for every function belonging to the Paley- Wiener space. Clearly, completeness in C [ - A, A ] implies completeness in Lp[ - A, A] for every value of p . It is often advantageous to permit repetitions among the An. Suppose, for example, that some A,, call it A, is repeated m times. In this case the understanding is that f(z), be it of the form (1) or (2), shall have a zero of multiplicity rn a t z = 1. It follows that either $ ( t ) or the differential dw(t) will be orthogonal to each of the functions

and hence all of these functions will be available for the approximation. Conversely, the orthogonality of &t) or dp(t) to all functions of this form implies that A is a zero of multiplicity rn of the corresponding entire function f(z). Infinite repetitions among the A, are excluded. For simplicity, we shall suppose in what follows that the A, are distinct. The reader will have no difficulty in reformulating all completeness theorems to include functions of the form (3). We begin with a closer look at the trigonometric system.

1

The Trigonometric System

We have seen that the trigonometric system {ein'}?= is complete in every Lp space (1 5 p < 00) over an interval of length 2n. In fact, if &t) is integrable on [ - R , R ] and

SIn

4(t)einfdt = 0

(n = 0, & I , k2,. . .),

then $(t) must vanish almost everywhere. Once a collection of functions is known to be complete, the next concern should be for economy-are there any superfluous terms'?

Definition. A system of complex exponentials (eiAn*}is said to beexact in

Sec. 11

113

The Trigonometric System

L p [- A, A ] or in C [ - A, A ] if it is complete, hut fails to he complete on the removul of uny one term. I f the system becomes exact when N terms are removed, then we say it has excess N ; if it becomes exact when N terms

are udjoined, then we say it hus deficiency N . As we shall see (Theorem 7), the particular functions e"" added or removed are arbitrary ; only their number is important.

Proposition 1. T h e trigonometric system is exact in L p [-n, n] for 1 5 p < x : in C[ -n, n] it hns deficiency 1. Proof. The first assertion is obvious-the trigonometric system is complete and its elements are mutually orthogonal. For the second assertion, argue as follows. It is clear that the trigonometric system is incomplete in C[ -n, n] since the only continuous functions that can be uniformly approximated by trigonometric polynomials are those for which f ( n )= f ( - 7 ~ ) . Let us adjoin the element eipf,where p is not an integer. If .f' E C[ -n, n ] , then for a suitable constant c the values of the function f(t) -

ceipf

at t = n and t = - n are the same. The new system is then complete, by virtue of the Weierstrass theorem on trigonometric approximation. Thus in C[ - 71, n] the trigonometric system has deficiency 1. I It is not difficult to show that if the system {eiAnr} is complete in L ' [ - A, A ] , then its deficiency in C[ - A, A ] cannot exceed 1 (see Problem 2).

Proposition 2. The frigonometric system is incomplete in L' over every inferoal of length yreuter than 2n. Proof. Indeed, given

I:

> 0, choose

&t) =

1:: 0,

t < -n It1

+E,

s ?r - c,

t>n-c,

114

The Completeness of Complex Exponentials

[Ch. 3

and put

f(z)

=

s"'

4(t)eiZr dt.

-n-e

A straightforward calculation shows that 4i

f ( z ) = - sin nz sin E Z , Z

and hence f ( n ) = 0 for every integer n. Since follows. I

E

was arbitrary, the result

Consider now the set of functions {einr},"= obtained from the trigonometric system by removing roughly "half" the terms. It s e e m plausible that the resulting set will be complete in L p over an interval of length n and over no larger interval. Surprisingly, this is not the case. The set { e ' " ' } ~ = is complete in Lp and in C over every interval of length less than 2n. This is a consequence of the following theorem of Carleman [1922].

Theorem 1. Let {A,} be a set qf positive real numbers and suppose that for some positive number A ,

Then the set {eiAnr}is complete in C [ - A, A ] .

Proof. We argue by contradiction. If the system were incomplete, then we could find a function o(t)of bounded variation on [ - A , A ] such that the entire function A

f(z) = { - A eizrd d t ) vanishes at each A,, but does not vanish identically. It is to be shown that condition (1) forces f(z) to have "too many" zeros, given the restriction that (2) imposes on its growth. The pertinent fact concerning the growth of f(z) is the obvious estimate

Sec.I]

115

The Trigonometric System

Applying Carleman's formula to j ( z ) in the right half-plane and then using (3), we obtain

1 (L I. < R

&) 5 & 1;($ S

A log R n

-

- $)2Ar

dr

+ O(1)

+ 0(1),

and hence

We complete the proof by showing that

Evidently, the right-hand side cannot exceed the left. If p is an arbitrary positive number belonging to the interval (0, l), then for each value of R ,

It follows that

and hence that

116

Letting

The Completeness of Complex Exponentials + 0,

[Ch. 3

we obtain (4).Conclusion

which is contrary to assumption. The contradiction proves the theorem. I Corollary. Proof.

The system {ein'}F= is complete in C [ - A, A] whenever A < n.

Since 1 N 1 -+l log N n = , n

__

1

as N + m ,

the result follows at once from Theorem 1. I The next result also follows readily from Theorem 1. The details are left to the reader.

Theorem 2. I f {A,,} is a set of positive real numbers for which n A lim inf - > -, n-m

An

7~

then the system { d i n t }is complete in C[ - A , A ] .

It is possible to replace Theorem 2 by much sharper results. In Section 6 it will be shown, for example, that if n A lim sup - > -, n-rm

4

71

then the conclusion of the theorem still holds.

Problems 1. Show that the deficiency of a system of complex exponentials { d i n t } in L p [- A , A] is a nondecreasing function of p . 2. Prove: if the system { e i i n ' } is complete in L ' [ - A , A ] , then its deficiency in C [ - A, A] cannot exceed 1. (Hint: Suppose that 1 does

Sec. 11

The Trigonometric System

117

not belong to { e i a n t ) The . set of all functions of the form

where j'E L ' [ - A, A ] and k is a constant, is a dense vector subspace of C [ - A , A ] . ) 3. Prove Theorem 2. 4. (Levinson) Let {A,,} be a set of positive real numbers and let n(t) denote the number of A,, in the interval [0, t ] . Then the system {eian'} is complete in C [ -nD,nD] if

(; +

limsup(lR n(t)Ji - Df 1

k)dt =

CO.

R-m

( H i n t : If { t i a n 'were } not complete, then we could find an entire function f'(z), zero for every A,,, but not identically zero, and expressible in the form

where o(t)is of bounded variation on [ -nD, nD]. Put g(z) = f ( z Z ) . By using Carleman's formula, a contradiction can be obtained in much the same way as in the proof of Theorem 1.) 5. Show that the criterion in Problem 4 is sharp in case n/A, exists. 6. Show that if

A,, = n

-

for n

n'"

=

2,3,4,. . .,

then the system { eiAn*}is complete in C [ - TE,n ] . 7. Let {A,,} be a sequence of positive real numbers for which lim sup I1-w

Is it true that

n ~

An

=

L.

118

2

The Completeness of Complex Exponentials

[Ch. 3

Exponentials Close t o the Trigonometric System

In this section we shall discuss the stability of the trigonometric system in Lp under small perturbations of the integers. Our goal is to show that if 1 < p < 00 and {A,,} is a sequence of real or complex numbers for which

II,, - nl

1 s 2P' -

n

=

0, +1, + 2 ,...)

then the system {e'".'} is complete in Lp[-n, n] (cf. Kadec's &theorem). This result is sharp in the sense that the constant 1/2p cannot be replaced by any larger number. As a measure of the density of a sequence of complex numbers {A,,}, we denote, as usual, by n(r) the number of points A,, inside the disk IzI 5 r , and we put N(r) =

j:

dt.

The following theorem is fundamental.

Theorem 3 (Levinson).

is complete in L p [-n, n], I < p < m, The set {eiAn.'}

whenever

r+m

P

Proof. Suppose to the contrary that the set {eiAnr}fails to be complete in LP[ - n,n] for some value of p . Then there exists a function 4 in L4[- n,n] such that the entire function

vanishes at each A,,, but does not vanish identically. We may suppose without loss of generality that the norm of 4 does not exceed 1. Let E be a small positive number. Then

Sec. 23

Exponentials Close to the Trigonometric System

119

and, by an application of Holder’s inequality, we see that lf(z)l

5 enlpllyl~l/P(e-”IYI+ A),

where

Clearly, I 0 as t: 0. If we now apply Jensen’s formula to f ( z ) and then use (I), we obtain -+

-+

N ( r ) 5 2r

-

1 -1ogr

+

P

logIe-crlsinel+ 21 d8

+ O(1)

(2)

since the R, form a part of all the zeros of j ’ ( z ) . Furthermore, if E is small enough and r is large enough, then the integrated term on the right-hand side of (2) can be made smaller than any preassigned negative number, and it follows that

P

r-ta,

The result now follows by contraposition. I

Theorem 4 (Levinson). !f’ 1 < p < io and {A,,] is a sequence of real or complex numbers ,for which

then the system (eiLnf) is complete in L p [ - n x ,n]. T h e constant 1/2p is the best possible.

Proof. Condition (3) implies that n(t) 2 1

(remember that

+2

[

[XI denotes

t -

~

whenever t > 1

;PI

the function “bracket x”). It follows that if

The Completeness of Complex Exponentials

120

[Ch. 3

r > 1, then

N(r) =

l;

1

22

?dt

t

+ 2[t

- 1/2p] dt t

dt

-

2

jl

t - 1/2p - [ t - 1/2p]

t

-

$

dt.

+

Since x - [x] - is periodic and has average value zero over each interval of length one, the last integral above remains bounded as r + 00 (see Problem 1). Accordingly, 1 N ( r ) 2 2r - -logr - c P

for some constant c and all values of r . Hence Theorem 3 applies, and the system {ei'n'} is complete in LJ"-n, n]. We complete the proof by showing that if E is an arbitrary positive number and

n

= 0,

then the set { e'""'} is not complete in L p [-71, n]. Put c = 1/2p E . It is to be shown that the function

+

4(t)= (cos + t ) 2 c -' sin i t is orthogonal to every element eiAn'( n = 0, f 1, f2,. . .); clearly, 4(t) belongs to Lq[ - 71, .]. Suppose first that n is positive. By writing s i n i t and cosit in complex form, we obtain

Sec. 23

Exponentials Close to the Trigonometric System

121

(by the binomial theorem) for n = 1 , 2 , 3 ,....

=O

This shows that & t ) is orthogonal to each of the functions eiAnf (n positive). A similar result holds when n is negative (take conjugates); the case n = 0 is obvious. I Remarks

1. Theorem 4 provides simple examples of sets that are complete in Lp[-n, n] but fail to be complete in Lr[-n, n] if r > p . 2. When p = I , the conclusion of the theorem is no longer valid. If

1

it

An

=

+ f,

n > 0,

0,

n

n - 7I ,

n<0,

=

0,

then the system {eiAnllis not complete in L ! [ - x , n]. Reason: The function sinft is orthogonal to every element of the set (verification is left to the reader). On the other hand, completeness in Lp, 1 < p < m, over a finite interval implies completeness in L ' , and hence the system {eiAn'}is complete in L' [ - n, n] whenever

Problems 1.

Show that the integrals

remain bounded as r

+

'x!

.

122

The Completeness of Complex Exponentials

2. Investigate the validity of Theorem 3 when p 3. (Levinson) If 1 < p < 00 and if 1

1

=

1 and when p

[Ch. 3 = KJ.

n = 0 , + 1 , + 2 ,...,

[A,,l~lnl+-+-N, 2P 2

then the set { eiAnr} is either complete in L p [- n,n] or else has deficiency at most N . 4. If {E,,};=,-, is a sequence of positive numbers such that x c , / ( n + 1) < a, then the system {e'"'}?, will be complete in L"[-n,n], 1 < p < co, whenever

3 A Counterexample Theorem 4 shows that the set of complex exponentials {eiAnl}is complete in L 2 [ - n,n] whenever

In the present section we shall prove that this assumption allows the system {eian'}to have an excess. This will serve to show that Kadec's ;-theorem is "best possible": the system {einnr}constitutes a basis for L2[-n, n] whenever every inis real and

but need not constitute a basis when L

Theorem 5.

=

i.

The set { e * i ( n - * ) r = : n 1 , 2 , 3 , .. .} is complete in L z [ -n, n].

= -1, (n = 1 , 2 , 3 , . . .). If the set {eiAn'} Proof. Put 1, = n - and L,, were incomplete in L 2 [ - n , n ] , then we could find a nontrivial entire function f ' ( z ) , expressible in the form

f(z)

=

l:n

$(t)eizrd t ,

Sec. 4

with

A Counterexample

123

4(t) in L 2 [ - TI,TI], such that f(An) = 0

for every An.

(1)

Furthermore, we may assume that

is complete in L 2 [ -71,711, conditions (1) (why?). Since the system { 1, eiAn'} and (2) determine 4(t)uniquely up to a set of measure zero. It is convenient to present the proof in three steps.

1. f ( z ) is an even function. It is sufficient to show that +(t)is even, i.e., that

Put

Since A-,l = -An, it follows that

F(A,)= 0

for every 1";

in addition, F(0) = 1. But $(r) is unique, and hence almost everywhere

This shows that

4(f)is even, as asserted.

Call the infinite product g(z) and set

124

The Completeness of Complex Exponentiats

[Ch. 3

Then @(z) is entire. We claim that @(z) is free of zeros. Argue as follows. If n(r) denotes the number of zeros of f(z) in the disk IzI 5 r , then by Jensen's formula,

Since

it follows that

1;

y d t 5 2r

+ O(1).

(3)

Let nl(r) and n2(r) denote the number of zeros of @(z) and g(z), respectively, within the disk ( z ( 5 r , so that

and hence

Since n 2 (t) = 2[t

+ $1, it follows just as in the proof of Theorem 4 that

{i

dt 2 2r

1 2

- - log r - O(1).

Combining (3)-(5), we see that n,(t) -dc 10

1 5 -1ogr

2

+ O(1).

But this is possible only if nl(t) is identically zero, i.e., if @(z)has no zeros. Since f(z) is of order no greater than one, Hadamard's factorization theorem justifies writing

S(z) = eA'g(z).

Sec. Jl

125

A Counterexample

But J'(z) and g(z) are both even, and hence A must vanish. Thus f

= 9.

3 . Define an entire function h(z) by setting

where the constant u is chosen so that h(0) = 1. Assertion for every An.

h(&) = 0

The method of proof is the same as that used for a similar purpose in the proof of Theorem 4.If n is positive, then

=

$ lim

(1

+ reit)-1/2einrdt

r-I-

= 0.

A similar result holds when n is negative. The proof of the theorem is now at hand. We conclude exactly as with f ( z ) that

and hence that f(z)

3

h(z).

But this is possible only if $(t)

=

a(c0s it)-

which does not belong to L 2 [-n, n]. We have arrived at a contradiction, and the result follows. I

126

The Completeness of Complex Exponentials

[Ch. 3

Problems 1. Show that the system { e * i ( " - * ) t : n= 1,2,3,. . .} is exact in L2[-z,

R].

( H i n t : The function

is entire of exponential type at most

R

and bounded on the real axis.)

2. Show that if g(z) is defined as in Problem 1, then

H i n t : g(z) =

r2(3 r($+ z)r(i- z)'

3. Is the system (eiicn+*)':n= 1,2,3, ...} exact in L2[-7r,7c]?

4

Some Intrinsic Properties of Sets of Complex Exponentials The close interplay between entire functions of the form

where w(t) is of bounded variation on the interval [ - A , A ] , and systems of complex exponentials {eiAnt}endows these systems with special intrinsic properties. One of the most useful states that the completeness of such a system is unaffected when one of the terms eiAtis replaced by a different term eiPt (different from those already present). The proof of this assertion is particularly transparent in the case of L ' [ - R , R ] . For if j ' ( z ) belongs to the Paley-Wiener space P,if

f(1,)= 0 and if

whenever An # 1

Sec. 41

Some Intrinsic Properties of Sets of Complex Exponentials

127

then the function

also belongs to P and satisfies

for every 1".

g(A,,) = 0

If (eiAn')is complete, then (E.,,) is a set of uniqueness for g(z), and so g(z) must vanish identically. It follows that f ( z ) must also vanish identically, and hence the new system (with eiA' replaced by eC') is also complete. For Lp spaces in general, the argument above must be modified. The essential ingredient is not the Paley-Wiener theorem, but the fact that if f ( z ) is the Fourier transform of an L p function and f ( p ) = 0, then g(z) is also the Fourier transform of an LP function. This is the content of the next theorem. Theorem 6. Let

where a ( [ )E L p [-n, n] and 1 S p < a.If f'(p) = 0 and

then there e.xi,sts 11 function O(t) in LP[ -n, n] such that

g(z) =

j:

n

The relation

holds almost etwrywhere

oil

[ - IT, n].

O(t)ei" d t .

128

The Completeness of Complex Exponentials

[Ch. 3

Proof. To motivate the proof, let us suppose that g(z) is in fact representable in the form (l), and then try to deduce (2). If (1) holds, then a(t)eizfdt

=-

b(t)eizfdt .

The trick in solving for P(t) is to transform each of these integrals by first rewriting eizras &zf

-

ei (z-p)f

eipf

and then integrating by parts. When this is done, the result is 1

with j l ( t )

= - ie-ia'

1'-

"

a(t)ei" dt

=

b(s)eiasds. It follows that

almost everywhere on [ - n,n], and so (almost everywhere)

To obtain ( 2 ) , differentiate both sides of this equation with respect to t . To prove the theorem, simply observe that all of the preceding steps are reversible; clearly, P ( t ) E LJ" -n, n]. I

Remark. A similar result holds when f ( z ) is of the form

and ~ ( tis) of bounded variation on [ -n, n], only now

Sec. 41

Some Intrinsic Properties of Sets of Complex Exponentials

129

with d/l(r) = dcr(t)

+ i(A

-

p)e-i"'

eipsdcr(s).

The following corollary is now immediate. Theorem 7. The completeness ofthesyhtem {e''p2']in L"[ - A, A ] , 1 5 p < E, or in C[ - A, A ] is unidJected if home A,,is replaced b y another number. For the remainder of this section, we shall restrict attention to uniform approximation over a given finite interval of the real axis. The reader will have no difficulty in verifying that all results remain valid for approximations in the L p metric. In order that an arbitrary sequence of continuous functions f l , f 2 .. f 3 , . . . defined on a closed interval [u, b ] be complete in the space C[a, b ] , it is necessary that e w r y element of the space be uniformly approximable by finite combinations of the f,,'s. For sequences consisting of complex exponential functions e""', the situation is dramatically simpler: such a system is a fortiori complete provided only that one other exponential function e'" can be so approximated. Theorem 8. The system { eiin') is complete in C[a, b ] if and only if i t s closed linear span contuins one other exponential function ei". Proof: The necessity is trivial. For the sufficiency, suppose that for some number ,i(different from every A,,), the function eiArcan be uniformly approximated on [u, h] by finite linear combinations of the elements eiAn' (n = I , 2,3,. . .). Then the constant function 1 can be uniformly approximated by finite combinations of the elements e"""-"' . Since I , - I is never zero, integration shows that all of the powers 1 , t , t 2 , 9 , .. .

can be so approximated. We conclude that for each polynomial p ( t ) the function e'"p(t)

can be uniformly approximated by finite linear combinations of elements from the set {ei""'). The proof is over: If J' is an arbitrary element of C[a, b ] , then the function e - i*'.f ( t )

130

The Completeness of Complex Exponentials

[Ch. 3

can be uniformly approximated on [a, b] by a polynomial, and the result follows. I

As a corollary we have the following theorem.

Theorem 9. Every incomplete set of complex exponentials in C [ a , b] i s minimal. (Recall that a set of vectors in a normed vector space is said to be minimal if each vector in the set lies outside the closed subspace spanned by the others .)

Definition. A set of vectors in a normed vector space is said to be linked if each vector in the set lies within the closed subspace spanned by the others. Theorem 10. Every set of complex exponentials in C[a, b] is either minimal or linked. Proof. If the set {ei"'} is not complete in C[a, b ] , then by virtue of Theorem 9 it is minimal, and there is nothing more to be said. If it is complete, then it is either minimal or else some element, say eiLk',belongs to the closed subspace spanned by the others. In the latter instance, the set { eiAn'} remains complete when the kth term is removed, and hence (by Theorem 7) when any term is removed. This shows that each element of the set { eiLn'} belongs to the closed subspace spanned by the others. I

Problems 1. (Paley-Wiener)

The system {eiAnt}is minimal in L2[ -n, n] if and only if there exists a nontrivial entire function f ( z ) of exponential type at most n, zero at every A,, and such that

2.

Let {A,} be an increasing sequence of positive real numbers. Show that in C[O, 11 the system {t"} is either minimal or linked depending on whether the series Xl/& is convergent or divergent.

Sec. 51

131

Stability

is complete in Lp[-n, 71.3, 1 < p < m,

3. Prove that the system (e'""')', whenever

5

Stability

We have seen that the completeness of the trigonometric system is preserved under sufficiently small perturbations of the integers. It is only natural to ask whether a similar assertion is true for sets of complex exponentials in general. Suppose, for example, that the set {eiAnf}is known to be complete in L 2 [-n, n]. Can we find a positive number E with the property that the set { ei""'j is also complete in L 2 [-n, n] whenever

The answer is no. Take it

- z 1,

n > 0,

Then the set of exponentials {&'} is complete in L 2 [-n, n] by Theorem 5. Choose c > 0 and put

+

in =

E,

A,,- E ,

n > 0, n < 0.

a.

Then \A,,- pnl = E , and if E is sufficiently small, then also sup,lp, - nl < It follows from Kadec's :-theorem that the set { 1, eiwnf}forms a (Riesz) basis for L 2 [ -n, n], and hence the removal of a single term (in particular, the element 1) leaves an incomplete set. One of the ways in which condition (1) may be strengthened is by requiring that

132

The Completeness of Complex Exponentials

and then imposing suitable restrictions on the take {c,) in 1'.

[Ch. 3

It is always sufficient to

E,.

Theorem 11. Let {A,} and { p , } be two sequences of real numbers ,for which m

n=

1

and let 1 p < a.~f{ei'nt} is complete in LP[- A , A ] , then complete in LP[ - A, A ] .

{dun'}

is also

Proof. Suppose to the contrary that { eiunr}is not complete in L p [- A, A ] . Then there exists an entire function f ( z ) , equal to zero for every p,, but not identically zero, and expressible in the form

with 4(t) in L4[ - A, A ] . , . .. Motivated by Theorem 6 , we define a sequence of functions &,$ J ~&, recursively by first setting do = I$ and then taking

SI

4,(t) = + n - l ( t ) + i(An - p , ~ e - ~ " n '

4,- l(s)eiflnsds

( n = 1,2, 3 , . . .). ( 2 )

A

Put

Then

by virtue of Theorem 6 , and consequently f,(z) the points

4

9 . .

Assertion:

{+,}

., 4, p,+

, p,+z 7

9

=0

whenever z is one of

. . ..

is a Cauchy sequence in L 4 [ - A A , A ] To . prove this,

Sec. 51

Stability

133

argue as follows. Since each P,~is real, ( 2 ) implies that

or

where

ZF.,< ,m.It follows that

Notice that the right-hand side of (5) shows that the sequence of n o r m I/ 4,11, I I = 1 , 2 , 3 , . . ., is bounded : indeed,

where C

=

n;=

(1

+ E , ) < m. Therefore, by virtue of (4),we have

cc since C c, < 00. This proves the assertion. which tends to zero as n, p Accordingly, there is a function I) in Lq[ - A , A ] such that 4, I) in Lq[ - A , A ] . Since no 4, is trivial (because of (3)), it follows that I) is not trivial (because of ( 5 ) ) . Put -+

-+

g(z) =

j:*

I)(t)e'"dt.

Since

we conclude that g(1,) = 0

for every A,,,

The Completeness of Complex Exponentials

134

[Ch. 3

and hence that {e'"'} is incomplete in Lq[ - A , A ] . The contradiction proves the theorem. I Theorem 11 provides a stability criterion for systems of complex exponentials { eiLn'}whenever the A, undergo a suitable "horizontal" displacement. Under a "vertical" displacement, much more can be said (at least in L 2 ) .

Theorem 12. Let {A,} and {p,} be two sequences of points lying in a fixed horizontal strip, and suppose that Red,, = Rep,

lf { e'""'} is complete in L2[- 71,711, then {eiPn'}is also complete in L 2 [- 71, 711. Proof. We may assume that neither set {A,} nor { p , } contains the point z = 0 (why?). Arguing by contradiction, we suppose that {eiPn'}is not complete in L z [- 71, 711, so that there exists an entire function f ( z ) ,zero for every p,,, but not identically zero, and expressible in the form

f(z) where

4~ L 2 [ - n n , n ] .In

=

$(t)eiz' dt,

addition, we may assume that f(0) = 1 (divide

f(z) by a suitable power of z and then invoke Theorem 6 ) . We define a sequence of functions f,,, f l ,f 2 , . , . as follows :

If n 2 1, then each f,,(z) belongs to the Paley-Wiener space P, assumes the value 1 at z = 0, and satisfies

f,,(A,)

=

0

(k

=

We are going to show that the norms

1,. . ., n).

Sec. 51

Stability

135

have a uniform upper bound for all values of n . For this purpose, write

with

4, in

L’[--.rr, n]. Since

(by definition), Theorem 6 implies that

Since { Imi,) and (Im p,) are both bounded, obvious estimates show that

for some constant A and all values of n , and hence

But the products

have a uniform upper bound for all values of n . Indeed,

for some constant B and k = 1 , 2 , 3 , . . ., and so

136

The Completeness of Complex Exponentials

[Ch. 3

Since f(z) is an entire function of order not exceeding 1, the exponent of convergence of its zeros is also no larger than 1, and so, in particular,

1/l&12 is also convergent, and hence that This shows that the series the infinite product in (6) is finite. It follows that

and hence also that

since the Fourier transform is an isometry. Since P is a Hilbert space, a subsequence of { f n } will converge weakly in P to an element g of P.This means that

for a suitable sequence of integers { n k }and all h in P. Choosing h to be the reproducing function at z , we find

for every z . In particular, g(0) = 1

and

g(1,) = 0

for n

=

1 , 2 , 3 , . ...

The proof is over. We have established the existence of a nontrivial function in P that vanishes at every A,,. Conclusion: The system {eiAn'}is not complete in L 2 [ -n, n]. The contradiction proves the theorem. I We have thereby obtained the following strengthening of Theorem 4.

Corollary. If{

A,,} is a sequence of complex numbers .for which

then the system { eiAnt} is complete in L z [ - n,n] .

Sec. s]

Density and the Completeness Radius

137

Problems 1.

(Redheffer) If

then the systems {eiArif) and {ei'lnf}have the same excess or deficiency in L p [ - A , A] for 1 S p < m. (Hint: If 411is defined as in the proof of Theorem 11, then

2.

3.

Does the conclusion of Problem 1 still hold when L p [- A, A ] is replaced by C[ - A, A ] ? Prove that the condition

is not strong enough to guarantee that the system of exponentials { ei&af 1, is a basis for L z [-n, n] (cf. Kadec's $-theorem).

,

6

Density and the Completeness Radius

For systems of complex exponentials { eianf ), the dichotomy between completeness and incompleteness, relative to a given space, is exceedingly complex. Thus, while the trigonometric system is incomplete in C [ -n, n ] , it becomes complete on the addition of a single arbitrary term cia'. An important measure of completeness, far less delicate than the algebraic notions of excess or deficiency, is that of the completeness radius. Definition. Let A = { A n } be u sequence of' real or complex numbers. The completeness radius of A is defined to be the number R ( A ) = sup(A : { eiAnf} is complete in C[ - A, A ] ) .

B y convention, R ( A ) = 0 if the set {eiAnf} f a i l s to he complete over uny interivd[ - A, A] und R( A) = oz if it is complete over every finite interval [ - A, A ] .

138

The Completeness of Complex Exponentials

[Ch. 3

It is not difficult to show that the value of R(A) remains the same if a finite number of points are removed from or adjoined to A. Moreover, the metric in the definition may be replaced by any L p metric, or by a variety of other topologies, and the value of R(A) does not change (see Problems 2 and 5). For a given sequence {A,}, the size of the corresponding completeness radius is closely related to the behavior of the ratios n(r)/ras r -,co. (Here, as usual, n(r) denotes the counting function of {A,}, the number of A, for , 5 r.) We shall say that {A,} has density D (0 S D 2 00) if which 111

If the A, are arranged in nondecreasing order of magnitude, then it follows readily from the definition that {A,} has density D if and only if . n lirn - = D. n-m

11nI

A sequence that possesses a density will be called a measurable sequence. The following theorem is simply a reformulation of Theorem 2 under the condition that {A,} is measurable.

Theorem 13. If {A,} is a sequence of positive numbers with density D and completeness radius R , then R 2 nD. Suppose now that {A,} is a (not necessarily measurable) sequence of It was shown by positive numbers, arranged so that 0 < A, < Az < Polya [1929] that of all the measurable sequences that contain {A,} as a subsequence, there is one whose density is a minimum. This smallest density is called the Pblya maximum density of the sequence {A,} ; it is equal to * a * .

It is clear that the Polya maximum density is at least as large as the “upper density” lim sup,, n(r)/r. We state without proof (see Problem 8) the following impressive generalization of Theorem 13,

Theorem 14 (Levinson). If {A,} is a sequence of positive numbers with Pdlya maximum density D and completeness radius R , then R 2 nD.

Sec. s]

139

Density and the Completeness Radius

Example. Consider the sequence 2k < m 5

{A,,]consisting of all integers m for which

zk + 2 k - ‘

(k = 1,2,3,. . .).

Then {A,,). consists of the numbers

--3,

5 , 6 , 9,10,11,12,

....

It is easy to see that ( A n } is a nonmeasurable sequence with Polya maximum density one, and so has completeness radius R = n. Indeed, the sequence has density one in every interval (2k,2k 2 k - 1 ) , k = 1 , 2 , 3 , .. .. It is precisely the density in these subintervals that determines the completeness properties of the system ( e i i n r } . Notice also that since

+

. . n 1 lim inf - = n+z An 2’ Theorem 2 implies only that R 2 n/2. The strength of Theorem 14 is thereby forcefully presented. In the early studies of nonharmonic Fourier series, it was well known that for a large class of sequences having density D the corresponding completeness radius is always equal to nD. Included in this category are those real sequences {A,,}“‘, that are symmetric ( I - , , = -A,,, n = 0, 1,2,. . .) and “close” to the integers in the sense that

/A,,- nl 5 L ,

n

=

0, f 1,

2, . . .

(Paley and Wiener [1934, p. 941). The conjecture by Schwartz [1943, p. 1301 that every real symmetric sequence having density D must also have R = nD therefore seemed eminently reasonable. The conjecture proved false. It was finally and dramatically refuted by Kahane [1958], who constructed a real symmetric sequence for which D = 0 and R = ix. Equally surprising was the result of Koosis [I9601 showing that there are subsets of the positive integers with D = 0 and R = n. Nevertheless, for a large class of sequences having density zero, one can conclude that the corresponding completeness radius must also be zero. Theorem 15. I f A = {A,,) is a sequence of r e d numbers such that C l/IJ.,,l < X , then R ( A ) = 0.

140

The Completeness of Complex Exponentials

[Ch. 3

The proof will require the following lemma.

Lemma.

Let

{A,} be a sequence of real numbers such m

thaf

r

then f ( 2 ) is a nontrivial entire function of exponential type at most xC, bounded by 1 on the real axis, and zero at every A,,. Proof. Observe first that since the function 1 - (sin x z ) / x z has a zero at the origin, there exists a constant B such that

I

sin xz

1 -~nz 15 Blzl

whenever

Iz1 5 1

Accordingly,

+ 00, it follows that the series Since \A,,\

sin (nz/A,) n= 1

xz/Ln

converges uniformly on each bounded subset of the plane (apply the Weierstrass M-Test). This shows that the infinite product in (1) defines an entire function f ( z ) that vanishes at every A,, but does not vanish identically. The simple estimate

implies that

Sec. s]

141

Density and the Completeness Radius

so that f(z) is of exponential type at most nC. When x is real, I(sinnnx)/ 7~x15 1, and hence I.f(x)l 5 1. I It is worth pointing out that we could not have taken f(z) to be the canonical product

n( UCI

P(z) =

3

1-- .

n= 1

Indeed, P(z) is of exponential type zero (by Theorem 2.7), and so cannot be bounded along the real axis (apply Bernstein's inequality). The proof of Theorem 15 is now readily established. Proof of Theorem 15. Let A be an arbitrary positive number. We are

going to show that there exists an entire function f ( z ) , zero for every A,, but not identically zero, and expressible in the form

with $(t) in C [ - A, A ] . This will show that the corresponding system of is incomplete in L p [- A , A ] for 1 5 p < 00 and hence exponentials {eiAnf} that R ( A ) = 0. Choose N so large that

and define

It follows from the lemma that g(z) is a nontrivial entire function of exponential type at most A , bounded on the real axis, and zero for z = I , ( n = N + l , N + 2 , N + 3 ,...). Let {zI,..., zM} b e a s e t o f M = N + 2 zeros of g(z), none of which belongs to the sequence {A,}, and form the rational function

The Completeness of Complex Exponentials

142

[Ch. 3

Evidently,

Put

Then f(z) is an entire function of exponential type at most A , zero at every I , , and not identically zero; in addition, f ( x ) ~ Ln ’ L2 along the real axis. By the Paley-Wiener theorem, we can write

jA

f(z)=

A

4(t)eiz‘

dt,

with 4 E L 2 [ - A , A ] . But the Fourier transform of an L’ function is continuous, and hence 4 E C[ - A, A ] . I The far-reaching study of the completeness radius culminated after more than thirty years in the celebrated work of Beurling and Malliavin [1967]. In a complete solution to the problem, these authors showed that for an arbitrary sequence A of real or complex numbers, the corresponding completeness radius R(A) is proportional to a suitably defined “density” of A. The proof, requiring new and deep properties of harmonic and entire functions, is well beyond the scope of the present work. Research on the broader question of completeness over an interval of length 2R(A) flourishes.

Problems 1. Let n(r) denote the counting function of a real increasing sequence I , , I,,&, . . .. Show that

n(r) n lim sup - = lim sup -. I-m r n-m In

Show also that a similar equality holds when limsup is replaced by lim inf. 2. Show that the value of the completeness radius does not change if

Sec. s]

3.

4.

5.

6. 7. 8.

9.

10.

Density and the Completeness Radius

143

in the definition we replace C [ - A , A ] by any one of the spaces L P [ - A AA, ] with 1 p < a. Show that the completeness radius of the sequence { I , } is equal to the greatest lower bound of positive numbers r for which there exists an entire function of exponential type T , bounded on the real axis, and zero for every I , . Prove that if the sequence (A,,} has completeness radius R , then the system of exponentials { eiLn'}has infinite excess in C[a, b ] whenever b - a < 2R and infinite deficiency whenever b - a > 2R. (a) Let ( I , ,I,, A 3 , . . . Ij and {pl, p,, p j r . . .} have completeness radii R , and R,, respectively. Show that { A l , pl, 1 2 ,p,,. . .} has completeness radius no larger than R , + R,. (b) Show that the value of R is unaffected by the addition or removal of any number of (real) terms A,, for which xl/lA,,l < cc. Give an example of a sequence for which the upper density is arbitrarily small and yet the Polya maximum density is infinite. For the example following Theorem 14, compute the value of the upper density of { I , , } . Prove Theorem 14 by assuming as known the following important but deep fact: the zeros of an entire function of exponential type r > 0 that is bounded on the real axis have an equal density 7/71 in both the right-hand and left-hand planes (Levinson [1940, Theorem VIII]). Show that if A = { A n } is a sequence of complex numbers for which

then R ( A ) = x. Determine the completeness radius of the sequence {2,3,5,7,. . .} consisting of all prime numbers.

This Page Intentionally Left Blank

4

INTERPOLATION AND BASES IN HILBERT SPACE

Interpolation and approximation often appear as two sides of the same coin; results about the one frequently imply results about the other. In the present chapter we shall exploit this duality to gain added information about Riesz bases in Hilbert space. This will then serve as the basis for a more penetrating analysis of nonharmonic Fourier series in L"-n,n]. Interpolation problems in an abstract Hilbert space have as their prototype the following classical trigonometric moment problem in L'[ - n, n] : &t)e-'"'dt

= cn

( n = 0, +1, + 2 , . . .).

Here the c,,'s are given complex numbers and a solution 4 in L'[ -n, n] is sought. As is well known, a solution will exist when and only when the scalar sequence (cn] belongs to 1'. The unique solution-unique because the trigonometric system is complete-is then given by

In its logical extension to an abstract Hilbert space H , the aforementioned trigonometric moment problem takes the form

Here f , , .f2,

f3,.

. . are vectors belonging to H , c I , c 2 , c 3 , .. . are scalars, 145

146

Interpolation and Bases in Hilbert Space

[Ch. 4

and f E H is to be found. Two special classes of sequences { f , ) will play a prominent role: (1) those for which m

whenever f E H (Bessel sequences), and (2) those for which the interpolation problem

is solvable whenever {c,} E l2 (Riesz-Fischer sequences). The first detailed investigation of sequences satisfying a “Bessel” property or a “Riesz-Fischer” property was made by Bari [1951]. Shared by every orthonormal basis, these two properties are-as we shall subsequently show-the critical ingredients of every Riesz basis.

1

Moment Sequences in Hilbert Space

Let H be a Hilbert space and { f , , f 2 , f 3 , . . .} a fixed but arbitrary sequence of vectors from H . By the nth moment of an element f E H we The sequence mean the inner product ( f , 1,).

is called the moment sequence of j , and the collection of all moment sequences is called the moment space of { f,}. The moment space will be denoted by M . It is clear that M is a vector space under pointwise addition and scalar multiplication. We shall be concerned with the following questions:

(i) When does a given sequence of scalars belong to M , i.e., under what conditions on {c,} does the moment problem

admit at least one solution? (ii) If there is a solution, is it unique? (iii) How can the solutions, if any, be recaptured from {c,,}?

Sec. 11

147

Moment Sequences in Hilbert Space

Of these questions, uniqueness alone is trivial: in order that the system of equations ( 1 ) shall admit at most one solution for every choice of the scalars cl, c2, cj, . . ., it is necessary and sufficient that { f,,} be complete. Proposition 1. Let { fl, f 2 , f 3 , . . .} he a sequence of oectors belonging to a Hilhert space H . l i .for some sequence of scalars {cI, c2, c 3 , .. .) the system (1) hus u solution f E H , then it has a unique solution of minimal norm. Proof. Let Y be the closed subspace of H spanned by the elements jl,f 2 , f 3 , . . .. Then Y contains at most one solution to the moment problem. For if f and y were solutions, each belonging to Y, then f - g would belong to both Y and Y l . Since Y n Y ' = { 0 } ,f = g . To see that Y contains at leust one solution to the moment problem, let f be an arbitrary solution in H and write

where

c1 E

unless f

Y and

= a,

E

Y'. Clearly (a, f,,)

it follows that

c1

=

c, for every n. Since

is the unique solution of minimal norm. I

Example 1 (Finite Interpolation). Let { f l , . . ., f , } be a linearly independent set of vectors from H . Then the finite moment problem

(f,fi) = ci

( i = 1, ..., n)

has at least one solution f E H for every choice of the scalars cl,. . ., c,. This follows easily from the fact that the Gram determinant

of an independent set is never zero. Indeed, the (unique) solution of minimal norm is then given explicitly by the determinantal formula

148

Interpolation and Bases in Hilbert Space

[Ch. 4

Example 2. The moment space of a Riesz basis is 1’. For if { f,} is a Riesz basis for a separable Hilbert space H , then it possesses a unique biorthogonal sequence ( g n } which, by duality, is also a Riesz basis. If { c , } E 12, then the series

n=l

converges to an element f~ H which clearly satisfies (1). Accordingly, M 3 12. On the other hand, the expansion

is valid for every element of H , so that { ( f ,j,)}E l2 and hence M c 1’. Thus M = 12.

Example 3. Let {A,} be a symmetric sequence ( I - , = -A,) of real numbers, and consider the following “generalized” trigonometric moment problem in L’[ -71,711:

s’

1 2n

-n

&l)e-iantdt = c,

( n = 0, +1, f 2 , . . .).

(2)

If {eian‘)is in some sense “close” to the trigonometric system, then (2) can be solved explicitly for 4 , We shall suppose that the system { e i a n t }forms a Riesz basis for Li[-71, n]. Then its moment space is Iz, and (2) admits a solution whenever { c , } is square-summable, and for these sequences only. If {g.} is the unique sequence in L 2 [-n, 711 biorthogonal to { e i a n t } then , the unique solution to (2) is given by the norm-convergent series

We shall determine the 9”’s explicitly. For this purpose the following theorem will be needed.

Theorem 1. Let {A,} be a symmetric sequence of real or complex numbers.

Sec. 13

149

Moment Sequences in Hilbert Space

r f the system of exponentials {eiAn'}is exact in L 2 [-n, n ] , then the product

converges to an entire function which belongs to the Paley- Wiener space.

Proof. If the system {eiAnt}is exact, then there exists an entire function f (z) which belongs to the Paley-Wiener space P such that f (A,) = 1 and f (A,,)= 0 for n # 0. By Theorem 2 . 1 8 we can write

f (z)

=

s'

1 2A

-x

cp(t)e'" d t ,

with cp E L 2 [-n, n]. Since {A,} is symmetric, cp( - t ) has the same orthogonality properties as cp(t). But cp(t) is unique (because {eiAnt}is complete) and so must be even. Hence f (z) is even. Now f (z) can vanish only at the An's with n # 0. For i f f (2) were zero for some other value, z = y , say, then the function

would also belong to P and would vanish at every A,. The system {eiAnt} would then be incomplete in L 2 [ -n, A ] , contrary to hypothesis. By Hadamard's factorization theorem

Since f (z) and the canonical product are both even, A

Returning now to Example 3, we write

n (1 $). m

C(Z) = z

-

n= 1

=0

and hence

150

Interpolation and Bases in Hilbert Space

[Ch. 4

For each integer n, the entire function G,(z) defined by

belongs to the Paley-Wiener space. Assertion: g, is the inverse Fourier transform of G,, i.e., for almost all t E [ -n, n], g,(t) =

Lrn

G,(x)eix'd x ,

where the integral is understood to be a limit in the mean (in the L2 sense). Call the right side h,(t). By the Paley-Wiener theorem, h,(t) = 0 almost everywhere outside [ -n, n]. By the Fourier inversion formula,

and therefore

's' 2n

h,(t)e-'lm' dt = h,,,,,. --11

Since the system { e i A n fhas } a unique biorthogonal sequence, we must have as asserted. By means of the Fourier isometry, the entire preceding discussion can be transformed into the Paley-Wiener space. The exponentials e'""' are transformed into the reproducing functions

g, = h, for every n,

g,(t) is transformed into

f(L,)

G,(z), while the moment problem itself becomes = c,

( n = 0, + I , ~ 2 ,. .), .

since f(1,)= ( f , K , ) . Here {c,} E '1 and f € P is to be found. The solution is immediate: f(z) is given by the norm-convergent series

Sec. 11

151

Moment Sequences in Hilbett Space

Moreover, since the expansion

is valid for every function f belonging to P and { G,} is a Riesz basis for P, it follows that

Thus, (3), with {c,) in 12, represents the most general function in P. Formula ( 3 ) is a simple example of a “generalized” Lagrange interpolation formula for an entire function assuming the values c, at the points A,,. We have obtained it under the assumption that {A,} is a symmetric sequence of real numbers for which the corresponding system of exponentials { eiin‘} constitutes a Riesz basis for t2[ -71, n]. In particular, the formula is valid whenever {A,} is real, symmetric, and

by virtue of Theorem 1.14. If we choose 1, = n, then G(z) = (sin nz)/n, and ( 3 ) reduces to the “cardinal series” for f ( z ) . The preceding examples indicate that there can be no simple solution to the general moment problem (J; f n ) = c,], n = 1,2,3,. . .. The following theorem provides a criterion for a solution which, although both necessary and sufficient, is in practice difficult to apply. Nevertheless, in the absence of additional information about the fit’s, it is frequently of use. Theorem 2. Let { f ,, f 2 , f 3 , . . .) be a sequence of vectors belonging to a Hilhert space H und { c , , c2, c 3 , .. .} a sequence of scalars. In order that the equations (f,f,)

=

(n = 1,2, 3 , . . .)

c,

sltall admit at least one solution f and sufjcient thut

E

H for which

f o r eoery finite sequence of scalars { a, .

I( 1 1 15 M , it

is necessary

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Interpolation and Bases in Hilbert Space

[Ch. 4

The necessity is trivial: if {a,} is an arbitrary finite sequence of scalars, then

Proof.

fl

For the sufficiency, let Y be the subspace spanned by the elements , f z , f 3 , . . . and define p on Y by setting A C a n f n ) = Cancn

for every finite sequence of scalars {a,,}. Condition (4)guarantees that p is defined unambiguously as a bounded linear functional on Y, with norm not exceeding M . It follows that we may extend p to all of H so that I p ( f ) l 5 M l l f l l . (Notice that the possibility of extending p in this way need not rely on the Hahn-Banach theorem: first extend p to the closure of Y “by continuity” and then define p ( f ) = 0 for f~ Y l . ) By the Riesz representation theorem, there is a vector f E H such that p(g) = (9, f) for every g E H . It follows that ( f , f , ) = cn ( n = 1 , 2 , 3 , . . .) and IJfll = IlPll M. I

s

Problems 1. Let X be an n-dimensional vector space, X ’ its algebraic dual, and { xl, . . ., xn1 and { f l, . . .,f n } linearly independent subsets of X and

X ’ , respectively. Prove that the generalized Gram determinant

2.

is not zero. Let { f , , f 2 , f 3 , . . .} be a linearly independent set of vectors belonging to a Hilbert space H, and suppose that the system

admits at least one solution. Let h be the (unique) solution of minimal norm. Let hn ( n = 1 , 2 , 3 , . . .) be the (unique) solution of minimal norm of the system

( f ,fi)

= ci

Show that h, + A in norm.

( i = 1,.

. ., n).

Sec. 2l

Bessel Sequences and Riesz-Fischer Sequences

153

Let {il, f 2 , fL3,. . .} be a basis for a Hilbert space H and M its corresponding moment space. Show that (f,,} is a Bessel basis if and only if M 3 I2 and a Hilhert basis if and only if M c /2. (For the definitions of Bessel and Hilbert bases, see Problem 4, Section 1.8.) 4. Prove that if f ( t ) E L2[0,11 and c, = sh t " f ( t ) d t , n = 0, 1,2,. . ., then {c,} €1'. ( H i n t : Show that the series Eio h,c, is convergent whenever {b,} E /2.) 5. Let {A,} be a symmetric sequence of real or complex numbers such that the system of exponentials {ei*n'}forms a Riesz basis for L 2 [ -n, n]. Show that the canonical product

3.

is an entire function of exponential type 71. 6 . Let xl,. . ., x,, be n linearly independent vectors from a normed vector space X and let cl,. . ., c, be arbitrary scalars, not all zero. Show that there exists an element f E X * such that f ( x i ) = ci

(i = 1,. . ., n).

(5)

7. With ( x l , .. ., x,} and {el,.. ., c,,} as in Problem 6, define A = ,?(el,. . ., c,,) by putting

where the infimum is over all scalars tl,. . ., 5, for which

(a) Show that the infimum is attained. (b) Show that the system ( 5 ) in Problem 6 has at least one solution f E X * for which 1 f 1 S L if and only if L 2 A. (c) Show that A is the minimum of the norms of all elements f E X * that satisfy (5).

2

Bessel Sequences and Riesz-Fischer Sequences

Bessel's inequality and the Riesz-Fischer theorem together embody one of the fundamental characteristics of an orthonormal basis: the moment

Interpolation and Bases in Hilbert Space

154

[Ch. 4

space is 1’. Because of their intrinsic importance, these properties have been abstracted.

Definition. A sequence { f , , f 2 , f 3 , , . .) of oectors belonging to a Hilbert space H is said to be a Bessel sequence if

for every element f E H . The sequence is called a Riesz-Fischer sequence if the moment problem

admits at least one solution f

EH

whenever {c,} E 1’.

Thus, { f,} is a Bessel sequence whenever its moment space is a subset of /’, while it is a Riesz-Fischer sequence if its moment space contains 1’. Proposition 2. Let {JZ> be a sequence of vectors belonging to a Hilbert space H . I]’ ifn> is a Bessel sequence, then there exists a constant M such that

f o r every f in H . If { f n } is a Riesz-Fischer sequence, then there exists a constant m such that (1) has at least one solution f satisfying

provided { c n ) E 1’. The proof of the proposition is left to the reader (see Problem 3). The numbers m and M appearing in the proposition are called hounds for the sequence. We state the following corollary separately in view of its importance for applications.

Sec. 2]

Bessel Sequences and Riesz-Fischer Sequences

155

Corollary. I f / j ; , ] is u Riesz- Fischer sequence in a Hilhert space, then the unit hall of' 1' can he intrrpoluterl in u uniJi)rmlji hounded way. The next theorem provides a fundamental characterization of Bessel sequences and Riesz -Fischer sequences in a Hilbert space.

Theorem 3. Let 1 f l , j i , j i . . . .) bL> u seyuence of vectors belonging to a Hilhert spuce H . Then ( i ) [ j ' ,, j i , j i . . . .) is u Bessel sequence with bound M if' and only if' the inequdity

(ii)

holds jbr eilery jinite sequence of scalars { cn>: j i , j i , . . .) is a Riesz-Fischer sequence with bound m only if'the inequality

if' and

holds for cwry jinite sequence qf sculurs { c,} . ) an arbitrary finite sequence of ProoJ (i) For the necessity, let ( c , ~ be scalars and put f = Cc,,fn. Then

11 .L 114

=

5

l(.Ly

j')12=

1 1

CnCf,

.Ln)12

cl(.l112 ClCL 1J2

5 MIIJ'I12CIc1,I23 by Proposition 2. Dividing by 1 j'[12,we find

For the sufficiency, observe first that

whenever ( c I I )E 12. If .f' is a fixed but arbitrary element of H , then for every

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Interpolation and Bases in Hilbert Space

[Ch. 4

square-summable sequence { c,} we have

It follows that

by the converse to Holder's inequality. (ii) For the necessity, fix an arbitrary finite sequence of scalars { c l , . . ., c N } and choose a , , a2,a 3 , .. . so that (1) a, = 0 if n > N , (2) = 1, and (3) Si,c,l2 = Ic,12. By Proposition 2, there is a constant m (independent of N and cl,. . ., c N ) such that the moment problem

z=l

Iz=,

has at least one solution f with Ilfll' 5 l/m. Applying Theorem 2, we find that

and the result follows. As for the sufficiency, it is enough to show that (3) has a solution f with 1 f 1 5 l/m whenever lan12S 1. Suppose then that {a,} belongs to the unit ball of 'I and let cl,. . ., cN be arbitray scalars. Since

z=

the result is, once again, a consequence of Theorem 2. I

Sec. 2]

157

Bessel Sequences and Riesz-Fischer Sequences

The criteria of Theorem 3 can be phrased more succinctly in the language of operators. Thus, { f i , f 2 , f 3 , . . .} is a Bessel sequence if and only if, given an arbitrary orthonormal basis { e l , e2, e3, . . .} for H, there exists a bounded linear operator T :H + H such that Ten = f ,

for n

1, 2, 3 , . . .;

=

it is a Riesz-Fischer sequence if and only if there exists a bounded linear operator S :H -+ H such that

Sf,,

=

e,,

for n

=

1,2, 3,. . ..

An equivalent formulation in terms of the Gram matrix

is also useful. In this setting, { f l , f 2 , f 3 , . . .} is a Bessel sequence with bound M if and only if A defines a bounded (self-adjoint) operator on 12, with norm not exceeding M . Reason: A is bounded by M if and only if the associated quadratic form

’.

is bounded by M , and this form equals 11 c i f i 1 Similarly, { f l , f 2 , . .} is a Riesz-Fischer sequence with bound m if and only if the “sections” of A are hounded from below by m. This means that

f3,.

for all n-tuples c = (cl,. . ., c,) and all n. Here the norms are Euclidean norms, and A, is the n x n matrix ( ( f i ,f j ) ) ; ,j = The reader will verify that this is equivalent to

for all n and c; it is also equivalent to 1 A,,-’

1 5 l/m

for all n.

Example 1. The sequence of powers { 1, t , t 2 , . . .} forms a Bessel sequence

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Interpolation and Bases in Hilbert Space

in L2[0,I]. In fact, if f € L 2 [ 0 ,13 and c, = j,!, t ” f ( t )dt ( n then

=

[Ch. 4

0, I , 2 , . . .),

f l

m

The constant n is the best possible. Reason: The corresponding Gram matrix A = ((ti, tj))Fj=O

+

is the “Hilbert matrix” (the i,jth entry is l / ( i j + l)), and so defines a bounded linear operator on 1’ of norm n (see Hardy, Littlewood, and Pblya [1952]; also see Problem 5). Suppose now that H is a functional Hilbert space, over a set S say, with reproducing kernel K , and let { I , , I,, A,,. . .} be a sequence of points in S . We shall take for the fn’s the normalized kernel functions K , / ) )K , ) ) , where K , ( z ) = K ( z , A,,). Then { f,,} is a Bessel sequence if

for every function f E H ; it is a Riesz-Fischer sequence if the “weighted” interpolation problem

f(&) JW,?I n )

= c,

(n = 1, 2, 3,. . .)

has a solution whenever {c,) E 12.

Example 2. As an illustration, let us take H to be the Hardy space HZ and { I , ,,I2, I,, . . .} a sequence of distinct points in the open unit disk IzI < 1 . Since the reproducing kernel of H2 (the Szego kernel) is given by K ( z , w ) = 1/(1 - zW),conditions (4) and ( 5 ) become, respectively, m

and f ( I , ) , / ~=~C,

( n = 1 , 2 , 3 , . . .).

(7)

Sec. 21

159

Bessel Sequences and Riesz-Fischer Sequences

We are going to show that (6) holds for every f E H2 provided that A, approaches the boundary of the unit disk exponentially. This means that there exists a constant c1 < 1 such that

It is enough to show that the Gram matrix A = (aij)TjI where

defines a bounded operator on 12. For this purpose the following simple form of Schur’s lemma will be needed.

Lemma 1. Let A = ( a i j ) be an infinite Hermitian matrix (aij = iiji), and suppose thut for some constant M rr

C laijJ5 M

( j = 1,2, 3,. . .).

i= 1

Then A defines a bounded linear operator on l2 of norm not exceeding M . Proof. It is sufficient to show that the associated quadratic form ( A x , x ) is bounded by M on 12, i.e., that

for every square-summable sequence x = { x i } If we write

1 U i j X i X j I = (1 a i j y1 X i l ) ( I aijl I X j l ) 1’2

and apply the Cauchy-Schwarz inequality twice, then (8) follows readily. I Let us return now to the Gram matrix under consideration. It follows by the assumption on (2,; that

and

Interpolation and Bases in Hilbett Space

160

[Ch. 4

Accordingly, for each j,

i

00

This establishes the Bessel condition (6) for all f in H2. Shapiro and Shields [1961] have shown that if 2, approaches the boundary of the unit disk exponentially, then the validity of (6) for every f E H2 is equivalent to the solvability of (7) for every sequence {c,} E 1’. There is a growing literature on interpolation problems in a wide variety of Banach spaces of analytic functions. For additional references, see the Notes (see, especially, Duren [1970], in which an account is given of Carleson’s proof of the “corona theorem”).

Problems 1. We shall say that a sequence of elements from a normed vector space

is uniformly minimal if for some E > 0 the distance from each element of the sequence to the closed linear span of the others is at least E . Show that every Riesz-Fischer sequence is uniformly minimal. Is the converse true? 2. Let {f,} and {g,} be complete biorthogonal sequences belonging to a Hilbert space H . Show that {f,} is a Bessel sequence if and only if {g,} is a Riesz-Fischer sequence. Is completeness necessary? 3. Prove Proposition 2. (Hint: If {f,} is a Bessel sequence, then the mapping T: H + 1’ defined by Tf = {(f, f,)} has a closed graph. If { f,} is a Riesz-Fischer sequence, then there is a natural mapping

4.

and this mapping has a closed graph.) Prove that a sequence of vectors { f.} in a Hilbert space H is a Riesz-

Sec. 2]

Bessel Sequences and Riesz-Fischer Sequences

161

Fischer sequence if and only if the eigenvalues of the sections A, ,fj))y, j = , n = 1,2, 3, . . . , are bounded away from zero. 5. (a) Prove Hilbert's inequality: if c, 2 0 (n = 0, 1,2,. . .), then

=

((fi,

Conclude that the Hilbert matrix defines a bounded linear operator on I' of norm not exceeding n. (Hint: Put f ( z ) = c,z". Using Cauchy's theorem, show that

This is the FejCr-Riesz inequality.) (b) By choosing f ( x ) = (1 - X)E-~'', E > 0, show that the constant 7c appearing in Example 1 is the best possible. Conclude that the norm of the Hilbert matrix must, in fact, be equal to n. 6 . Show that the Hilbert matrix is not invertible. 7. Show that the system (7) is solvable (with ~ E H ' )for every sequence {c,} E I' if

I, = 1

-

2-"

(n = 1,2,3,.. .).

( H i n t : The Gram matrix A = (aij), where aij = (&/[I Kill, Kj/II Kjll), is invertible.) 8. (Interpolation in B,) Let T be the mapping of B, to I" defined by

(a) Show that T maps B, onto I" if T > 7 c . ( H i n t : If E > 0 and if {c,} E I", then the series

c a'

S(Z)

=

c,

sin n(z - n ) sin E(Z - n) n(z - n) E ( Z - n)

converges in B, .) (b) Show that T does not map B, onto I". {c,) defined by

( H i n t : The sequence

if n S 0

cannot be interpolated (cf. Problem 4,Section 2.5, Part Two).)

Interpolation and Eases in Hilbert Space

162

3

[Ch. 4

Applications to Systems of Complex Exponentials

In Section 2 we considered Bessel sequences and Riesz-Fischer sequences in an abstract Hilbert space, and we showed that these properties are equivalent to a number of others. In the present section we shall apply these results to systems of complex exponentials {eianf}in L 2 [ - A, A ] . As a result, we shall obtain added information about closure and the completeness radius.

Theorem 4. I f { A n } is a separated sequence of’ real numbers, then the system { e i a n f }forms a Bessel sequence in L 2 [ - A, A ] for every positive number A . I f Elcn12 < m, then the series

C cneianf n

converges in the mean to an element of L 2 [- A, A ] . Proof. If

4 E L 2 [ - A, A ] , then the inner product an = ( + ( t ) ,e’””‘

is just the value f

(A,) of the entire function

Since f ( z ) is of exponential type (at most A ) and belongs to L2 along the real axis, we have C l a n J 2=

Clf(Jn)l2 < a n

n

(by Theorem 2.17). This proves the first statement. The second follows from the first by Theorem 3 . I If the separation of the 1,’s is great enough, then (eiant}is also a RieszFischer sequence.

Theorem 5. If { A,} is a separated sequence of real numbers such that n

An+,

-

An

L Y > -A

( n = 0, +1, +2,.. .),

then the system {eianf}is a Riesz-Fischer sequence in L 2 [- A, A].

163

Sec. 3]

Applications to Systems of Complex Exponentials

Proof.

Without loss of generality, we may assume that A = x , so that all finite sequences (ell) of scalars,

1) > 1. It is to be shown that for

where

-$)>o.

m = 2II ( ,

Let { c l t } be such a sequence and put

If k(r) is any function integrable over the real axis and

then

If we choose

then K(x)=

4 cos I I X 1 - 4x2' ~

and n:[

I./wl2

dr 2

1WA,,]

- A~)(.~C,,-

m. it

Let S, denote that part of the sum for which rn part. Clearly,

=

n and S, the remaining

164

Interpolation and Bases in Hilbert Space

[Ch. 4

Since K ( x ) is even and since 21crnZnlS 1crnI2+ IcnI2, there is a constant 8 such that 181 5 1 and

L

rnfn

n

rn

(the prime indicates omission of the term rn

= n).

Since

for rn # n , we have

Hence

and the theorem follows. I The theorem is sharp in the sense that if y = n / A , then the conclusion no longer holds. The following example is by now a familiar one. Example. The system of exponentials { e * i ( n - * ) f :=n 1 , 2 , 3 , . . .} is not a Riesz-Fischer sequence in L 2 [- n,n]. Write A,, = n - and = -An ( n = 1, 2, 3 , . . .). Let g(z) = l ~ = , , a n z n be analytic for IzI < 1 and define

a

f(t) =

+

eiA'g(reif) e-iA'g(re-if),

where 0 < r < 1 and A > 0. Then m

W

Sec.33

Applications to Systems of Complex Exponentials

165

and both series converge uniformly in t for fixed values of r and 1.If we take = i, then f ' ( t ) is of the form

A

with A,,+,- A,, 2 I . If the system of exponentials {eiAn'}were in fact a Riesz-Fischer sequence in L 2 [ - n , n ] , then we could find a positive constant m such that

Since a'

cIcn12 = 2

1lan12r2"= n=0

n

s'

(g(rei')12d t ,

-n

this would imply

for every r (0 < r < 1). Take g ( z ) = (1

+

z)-l'2

(the principal value is to be chosen). We leave it to the reader to show that for this choice of g , the integral on the right in (1) remains bounded as r -P 1, while that on the left is asymptotically - 2 x log(1 - r). But this is absurd, and hence {eidn'}is not a Riesz-Fischer sequence in L 2 [ - x , n]. The following result is an almost immediate consequence of Theorem 5.

Theorem 6 . If' {A,,}is a separafed sequence of real numbers such that x

A,,+, - A n 2 y > A

( n = 0, +1, f 2 ,...),

then the system {eidn')has infinite deficiency in L2[ - A, A ] . Proof. Let N be a fixed but arbitrary positive integer. If K is large enough,

interpolation and Bases in Hilbert Space

166

[Ch.4

then we can replace

so that the resulting sequence, relabeled {p,,},satisfies

By Theorem 5 there is a function

4 E L 2 [ - A, A ]

c$(t)e-iSnfdt

=

1 0

such that

if n = 0, if n # 0.

Thus the system {eiSn‘: n # 0) is incomplete in L 2 [- A , A ] , and we conclude by Theorem 3.7 that the deficiency of {eian‘}in L 2 [- A , A ] is at least N . Since N was arbitrary, the result follows. I Corollary. Every sequence {A,,}?m of real numbers ,for which

has completeness radius zero.

Problems 1.

Let {A,,} be a sequence of real numbers such that A n + l - A,, 1 1 for all n. Theorems 3 and 4 imply that there is a constant M such that

s MCIC,,J~

IICc,,eianf[12

whenever {c,) ~ 1 (The ~ .norm is in L 2[ - n , n ] . ) Show that M must be at least 2. 2. Give an “elementary” proof of Theorem 4 by using the same method used to prove Theorem 5 . : n = 1 , 2 , 3 , . . .} is not a Riesz-Fischer 3. Prove that the system {e*icn+t’‘ sequence in L2[ - n, n ] .

Sec. 41

The Moment Space and Its Relation to Equivalent Sequences

167

4.

Every basis je’””‘) for L 2 [ - A A , A ]such that the exponents A, are all real is a Hilbert basis. (Hint: Show that {A,} is separated. For the definition of a Hilbert basis, see Problem 4,Section 1.8.)

4

The Moment Space and Its Relation to Equivalent Sequences

Let { f1, f 2 , j i , . . .) be a sequence of vectors belonging to a separable Hilbert space. In what ways are the properties of the sequence reflected in its moment space? We have seen, for example, that if the sequence is a Riesz basis, then its moment space coincides with 1’. The converse, of course, is false: the removal of a single vector from a Riesz basis leaves an incomplete set whose moment space is again 1’. The goal of the present section is to show that the converse is true provided that { f l , f 2 , f 3 , . . .} is complete: a complete sequence of vectors belonging to a Hilbert space is a Riesz basis if and only if its moment space coincides with 1’. We begin by showing that the moment space characterizes complete sequences up to “equivalence”. Two sequences of vectors { f l , f 2 , f 3 , . . .) and {gl, g2, g 3 , . . .} belonging to a Hilbert space H are said to be equivalent if there exists a bounded = g,, ( n = 1,2,3,. . .). invertible operator T : H + H such that

rf,

Theorem 7. Two complete sequences of vectors belonging to a separable Hilbert space are equivalent if and only if they have the same moment space. Proof. Let { f l , f,, j ; , . . .) and (gl, g2, g 3 , .. .) be complete sequences in H and suppose that there exists a bounded invertible operator T on H such that Tf, = g, ( n = 1,2,3,. . .). It is to be shown that the system of equations

admits a (unique) solution f , whenever g is given, and a (unique) solution whenever ,f is given. Evidently this is true if

g,

and the condition is necessary. Suppose now that { f ! , f 2 , f 3 , . . .} and {gl, g, g 3 , .. .} have the same moment space. Then for each g in H there corresponds a unique vector

168

Interpolation and Bases in Hilbert Space

[Ch. 4

f such that (1) holds. We are going to show that this implies the existence of a bounded linear operator T : H --+ H such that Tfn = gn

(n = 1,2,3,. . .).

Let Y be the linear span of the fn's. I f f € Y , with f =

(2) cnfn, define

Tf by putting

n=l

We first show that T is well-defined. If it were not, then we could find scalars cl,. . ., cN such that

n= 1

n= 1

z=

Choosing g = cngn and then taking f to be the corresponding solution to ( l ) , we find

\

n=l

)

n=l

n=1

which is absurd. Thus T is well-defined. It is clear that T is linear and that (2) holds. To show that T is bounded, define a function p on H by writing

where f is the unique solution to (1). We claim that p is a bounded linear operator. Linearity is easy: if hl and h, are vectors in H and if a, and a2 are scalars, then for every n

Sec.41

The Moment Space and Its Relation to Equivalent Sequences

169

Since { f , } is complete, it follows that

and p is linear. Suppose now that { h , } is a sequence in H such that

h,+h

p(hn)+k.

and

Then

for all i, and hence by the definition of p we have k = p(h). Thus p is bounded by the closed graph theorem. Let f be an element in Y. Then for every g E H ,

Thus T is bounded on Y . Since Y is a dense subspace, we may extend T to be a bounded linear operator on all of H . Similarly, since (1) can be solved uniquely for g, given f, there exists a bounded linear operator S : H + H such that S g , = f,

(n

=

1 , 2 , 3 , . . .).

(3)

The proof is over. By combining (2) and (3), we see that ST = I = TS, since both { f,,} and {g,} are complete. Therefore, T is invertible, and { f , } and {g,} are equivalent bases for H . I The following important corollary is now immediate.

Theorem 8. A complete sequence of' vectors in a separable Hilbert space is a Riesz basis if and only if its moment space is equal to 1 2 .

Problems 1. Deduce Theorem 8 from Theorem 3. 2. Let { 1,)and {g,,} be two complete sequences of elements from a sep-

170

Interpolation and Bases in Hilbert Space

[Ch. 4

I:=,

arable Hilbert space H , and suppose that c,f, is convergent if and only if c,g, is convergent. Does it follow that { f , ) and { g,) are equivalent (cf. Theorem 1.7)? 3. Let { f l , f 2 , f 3 , .. .} be a complete sequence of nonzero vectors in a separable Hilbert space H . Prove that { f l , f z r f3,. . .) is a basis for H if and only if the moment problem

En"=,

admits a (unique) solution f for every sequence {cl, c 2 ,c 3 , .. .) satisfying the following condition: a,c, is convergent whenever a,f, is convergent.

5

Interpolation in the Paley-Wiener Space: Functions of Sine Type

Theorem 8 has far-reaching implications for the study of nonharmonic Fourier series in L 2 [ - n , n ] . Thus for a system

of complex exponentials to form a Riesz basis for L 2 [- n,n] it is necessary and sufficient that the trigonometric moment problem

2n

s'

4(t)e-iAnf dt

= c,

( n = 1,2,3,. . .)

-n

(1)

admit a unique solution 4 E L 2 [ - n , n] for each square-summable sequence {c,} of scalars, and for these sequences only. By virtue of the isomorphism between L z [-n, n] and the Paley-Wiener space, (1) is equivalent to the interpolation problem f(&)

= c,

( n = 1,2,3,. . .),

(2)

where f E P. Since P is richly endowed both with a Hilbert space structure and with a distinguished class of elements, problem (2) is often more readily solved than problem (1). In what follows we shall exploit this duality to great advantage.

Sec. 51

Interpolation in the Paley-Wiener Space: Functions of Sine Type

171

Definition. A sequence { A l , I,,, AJ,. . .) of real or complex numbers is said to he un interpolating sequence for P if the set o j all sequences

where f ranges over P, coincides with 12. I f , in uddition, the system (elin‘) is complete in L 2 [ - n, n ] , then ( 2 )has exuctly one solution, provided { c,} E 12, and in this case MY shall call {A,,A.2, A.3, . . .} a complete interpolating sequence. A complete interpolating sequence is clearly “maximal” in the sense that it is not contained in any larger interpolating sequence. The converse is also true (Problem 1). In the present setting, Theorem 8 takes the following form.

Theorem 9. A systvm [ei*n‘)of’ complex exponentials is a Riesz basis for L 2 [ -n, n] if and only if’ {A,) is a complete interpolating sequence for P.

Application :Functions of Sine Type

That the zeros of sin nz appear in the natural basis for L z [-n, n] is not entirely fortuitous. In fact, sinnz is but one-the prototype, to be sureof a large class of entire functions of exponential type whose zeros {A,} give rise to bases {din‘}of complex exponentials.

Definition. An entire function j ( z ) of exponential type rc is said to be of sine type I f ‘ (i) the zeros of f ( z ) are separated?, and (ii) there exisr positive constants A, B, and H such that

whenever x and y are real and lyl 2 H . According to the definition, a function of sine type is bounded on the real axis and so must have infinitely many zeros. The zeros are all simple and lie in a strip parallel to the real axis.

t This requirement is sometimes omitted from the definition

172

Interpolation and Bases in Hilbert Space

[Ch. 4

A large class of examples of entire functions that satisfy part (ii) of the definition is provided by the Fourier-Stieltjes integrals

SI.

eiz' d o ( t ) ,

where w(t) is of bounded variation on the interval [ - n, n] and has a jump discontinuity at each endpoint (Problem 3). Clearly, sin nz is of this form: simply choose w(t) = 0 for --n < t < x and o(k n) = 1/2i. We shall establish the following impressive theorem.

Theorem 10. If {A,} is the set of zeros ofa function of sine type, then the system { eiAnt}is a Riesz basis for L 2 [ -n, n ] . The proof will require several preliminary results. The first is a strengthening of condition (3) inside the domain lyl < H .

Lemma 2. Let f ( z ) be an entire function of exponential type n and suppose that (3) holds whenever lyl is sufficiently large. For each E > 0 there corresponds a number m > 0 such that

outside the circles of radius E centered at the zeros of f (z). Proof. Let A,, A 2 , A,, . .. be the zeros of f ( z ) . I f the lemma is false, then there exist positive constants E and H and a sequence z , , z2, z,, . . . of points that lie inside the horizontal strip lyl< H , but outside the circles Iz - A,[ = E ( n = 1,2,3,. . .) such that lim f ( z , )

= 0.

n+m

+

Write z, = x, iy,, with x, and y, real. Since sup,Iy,I < 00, we may suppose without loss of generality that limny, exists; call it yo. Next, define a sequence f i ( z ) ,f 2 ( z ) ,f 3 ( z ) ,. . . of entire functions by writing

f,(4 = f ( z

+ x,),

n = 1,2,3,. . ..

Since f ( z ) is bounded in the horizontal strip R

=

{z:IImzl < H

+

E},

the

Sec. 51

Interpolation in the Paley-Wiener Space: Functions of Sine Type

173

sequence { f,(z)) is uniformly bounded there and so forms a normal family in 0. Accordingly, there is a subsequence that converges uniformly on every compact subset of SZ to a limit function f o ( z ) . To simplify the notation, we shall continue to denote this subsequence by ( f , ( z ) } . Since f,(iy,)+O

as n - t c o ,

it follows that

fo(iYo) = 0. Since fo(z) does not vanish identically (indeed, If,(iH)I 2 AenH for n = 1, 2 , 3 , . . .), Hurwitz’s theorem implies that all but a finite number of the f,’s ( n = 1 , 2 , 3 , . . .) must have a zero inside the disk D = {z:Iz - iy,l < 4 2 ) . Say f,(w,,) = 0

with

W , E D and n > no.

If we set

+ x,

y,, = w,

then f(y,)

=

for n > n o ,

0, so that each y, must be one of the A’s. But

E E < - + - = E

2

2

as soon as n is sufficiently large, and this contradicts the choice of z , . The contradiction proves the lemma. I

Corollary 1. If f ( z ) is a function of sine type and 11,I,, A 3 , . . . are its zeros, then inflf’(A,)l > 0. n

Proof. Choose

E

> 0 so small that

[A,,- Am/ > 28

whenever n # m.

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Interpolation and Bases in Hilbert Space

[Ch. 4

Then the disks D,:Iz - A,\ S E ( n = 1 , 2 , 3 , .. .) are nonoverlapping, and each function f(z)/(z - A,) is analytic and free of zeros inside D,. If m > 0 is chosen in accordance with Lemma 2, then

and the result follows from the minimum principle for harmonic functions. I

Corollary 2. I f {A,} is the set of zeros of a function of sine type, then the system { eiAn‘} is complete in L ~-n, [ n]. Proof. Let f ( z ) be a function of sine type with zeros A,, A 2 , I , , . . .. It is sufficient to show that {A,} is a set of uniqueness for P. Suppose therefore that g(z) E P and g(A,) = 0 for n = 1, 2, 3, . . .. Then the function

is entire. By the corollary to Theorem 2.18,

and hence by Lemma 2, 4(z) + 0 as IzI + co,outside a system of (nonoverlapping) circles Iz - Anl = E ( n = 1 , 2 , 3 , .. .). The maximum principle guarantees that &z) + 0 inside the excluded circles as well, and therefore 4(z) must vanish identically. Hence g(z) vanishes identically, and {A,} is a set of uniqueness for P . I Let D be the upper half-plane, I m z > 0. We shall make use of the space H 2 ( D )consisting of all complex-valued functions f(z) that are analytic throughout D and such that

The following facts about H 2 ( D ) are well known (see Duren [1970, Chap. 111 and Problem 6 ) . 1 . If f~H 2 ( D ) ,then the boundary fitnction f ( x ) = lim f ( x Y+O

+ iy)

Sec. 51

Interpolation in the Paley-Wiener Space: Functions of Sine Type

175

exists almost everywhere, and

It follows that an inner product on H 2 ( D ) is defined by setting

Endowed with this inner product, H 2 ( D ) is a separable Hilbert space. 2. Every function J’ in H 2 ( D ) can be recovered from its boundary values by means of a Cauchy integral:

f(z)

=-

27ti

ja’ ~

-11,

j”‘) d t ,

t -z

I m z > 0.

3 . If ( A 1 , A 2 , A 3 , . . .) is a separated sequence of points lying in the horizontal strip

then

for some constant B and every element j’ E H 2 ( D ) . We are now ready to prove Theorem 10. Proof of Theorem 10. Let g(z) be a function of sine type with zeros 2 I , A 2 , i3,. . .. By Theorem 9 it suffices to show that {A,,) is a complete interpolating sequence for the Paley-Wiener space. This means that

(i) the interpolation problem f(A,)

= C,

(n = 1 , 2 , 3 , . . .)

has a solution , ~ E whenever P clc,,12 < 00, (ii) I.f(A,,)l2 < rx, whenever J’EP,and (iii) the system { e’””) is complete in L 2 [ -n, n].

c;=,

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Interpolation and Bases in Hilbert Space

[Ch. 4

Of these three statements, only the first requires proof: the second is valid because {A,} is separated and sup,IIm A,( co,while the third follows from Corollary 2. Let {c,} belong to 12. The most natural choice for a function that assumes the value c, at the point A, is the Lagrange interpolation series

-=

We are going to show that the series converges in the norm of P, and therefore uniformly in every horizontal strip; accordingly, f E P and f ( A , ) = c, (n = 1,2,3,. . .). Let m and n be arbitrary but fixed positive integers, with rn < n, and write

Then G(z)E P, and we have only to show that 1 G 1 -P 0 as m,n -+ Begin by choosing positive constants A, B, and H such that

for all values of x and lyl 2 H , and put

h(z) =

q z g(z

+ 2iH) + 2iH) .

Straightforward calculations show that h(z)E H2(D), and hence

(

- Be4nH I =

Be4nHIIh 1

Ih(x) I

')

1'2

(by property 1).

We estimate the norm of h as follows.

00.

Sec. 51

Interpolation in the Paley-Wiener Space: Functions of Sine Type

177

Choose cp in the unit sphere of H 2 ( D ) such that 1 h 1 = (h, cp). Then

J-

By Corollary 1 we may choose E > 0 so small that lg’(&)l 2 c for k = 1,2, 3 , . . ., and by property 3 we may choose B so large that the second sum on the right is at most B2. It follows that

and hence

where C is independent of m and n. Since { c,} E 12, the right side approaches zero as m,n m. Thus 1 ) G 1 0. I --+

--+

Problems 1. Prove that a maximal interpolating sequence for P is complete. 2. Let {A,,) be an interpolating sequence for P and f,(z) the unique element of P of minimal norm for which f,(l,,,) = b,,,,,.Show that if { c,} E I z , then the series

converges in P to a function f ( z ) for which f ( l , ) = c, for every n. 3. (Sedleckii) Let

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Interpolation and Bases in Hilbert Space

where w(t) is of bounded variation on [ -n, n] and has a jump at t (a) Prove that there exist positive constants A and H such that

[Ch. 4 = n.

for all values of x and y < - H . ( H i n t : Let h be the value of the jump of w(t) at t = n. Given E > 0, choose S > 0 so that the total variation of w(t) on the interval [n - 6, n) does not exceed E . Then

If y < 0, then

where I/ is the total variation of w(t) on [ -n, n].) (b) Prove that if (4) holds for a sequence of points iy, of the imaginary axis such that yn + - 00, then w( t ) has a jump at t = n. 4. Prove or disprove: Two functions of sine type with the same set of zeros can differ at most by a multiplicative constant. 5. Show that if f ( z ) E P,then f ( z ) e i n zE H z ( D ) . 6 . Establish property 3 for functions belonging to H 2 ( D ) . ( H i n t : Choose E > 0 so small that the disks D,:(z - znI S E are disjoint and lie in the upper half-plane, Im z > 0. Then

7. (Interpolation in E;) Let {I.,} be the zeros of a function g ( z ) of sine type, 1 < p 00, and { c n } a sequence in P. (a) Show that the series

-=

converges in Ef: to a function f ( z ) for which f(I.,) = c, for all n. (For the definition of E;, see Problem 5 of Section 2.3, Part Two.)

Sec. s]

Interpolation in the Paley-Wiener Space: Stability

(b) Take g(z) = sinnz. Exhibit sequences in the equations

f ( n ) = c,,

179 1P

( p = 1 , a)for which

( n = 0, & 1, +2,. . .)

have no corresponding solution in EE. ( H i n t : See Problem 8, Section 2. By definition, E,"' = Bn.)

6

Interpolation in the Paley-Wiener Space: Stability

By definition, a sequence { A l , A,, A 3 , . . .} of scalars is an interpolating sequence for the Paley- Wiener space provided that (i) the equations ,f(A,,) = c, ( n = I, 2,3,. . .) have at least one solution f E P for each se. first condiquence { c n ) in Iz, and (ii) ~ l f ' ( A , , ) 1 2< cc for every ~ E PThe tion is of course the more difficult one to verify; given the first, the second frequently comes gratis. Proposition 3. Let If the equations

A,,A,, A 3 , . . . f(A,,) = c,,

lie in a strip parallel to the real axis. ( n = I, 2, 3 , . . .)

admit at least one solution f ' E P for each square-surnmable sequence {c,,} of' scalurs, then (A,,} i s separated, and so

Proof: Since P is a functional Hilbert space, there is a sequence K , , K , , K , , . . . of elements of P such that

for n = 1,2,3,. . . and every f' E P. Indeed, if K is the reproducing kernel of P. then

K,(z)

=

K ( z , A,,)=

sin n(z - I,,) n(z - I,,) .

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Interpolation and Bases in Hilbert Space

[Ch. 4

By assumption, { K , , } is a Riesz-Fischer sequence in P, and hence the unit ball of I’ can be interpolated in a uniformly bounded way. In particular, there exist a constant M and a sequence fi,f2,f 3 , . . . of elements of P such that fn(Am)

= 6mn

and

1 1 5 M* f n

Choose H so large that IImA,I 5 H for every n. If rn # n, then

and so

where the supremum is taken over all points z on the line segment joining A,, and I,,. For these points, IIrnzl S H , and it follows from known properties of P that

Thus 1 5 MneXHI1,- A,,,[

whenever n # m,

and {A,,} is separated. Since sup,,IImA,I < co, the Remark following The< 00 whenever f e P . I orem 2.17 shows that

Clf(A,,)l’

Corollary 1. The points of an interpolating sequence lie in a strip parallel to the real axis and are separated. Proof. Let {A,,} be an interpolating sequence for P, and let { K , } be the corresponding sequence of kernel functions (1). By assumption, { K , } is a Bessel sequence in P, and so by Proposition 2

for some constant B and every function f E P. It follows readily from this that

)I 1c,,eiLnt 1)’ 5 B 1I c, 1’

Sec. s]

Interpolation in the Paley-Wiener Space: Stability

181

whenever (c,} E l2 (first observe that J J ~ C , , K 5 , , JB\X~\ C , \ ~and , then use the Fourier isometry). We conclude that

and hence that

This proves the first assertion; the second follows from Proposition 3. I Corollary 2. If the system of exponentials {eiAnf}is a Riesz basis for L 2 [-n, n], then the points A, lie in a strip parallel to the real axis and are separated. The goal of the present section is to show that interpolating sequences in the Paley- Wiener space are stable under sufficiently small perturbations of their elements. It is the first step towards a general stability criterion for Riesz bases of complex exponentials (see Section 8). We begin by proving two lemmas, which are of interest in their own right.

Lemma 3. Let {A1, A 2 , A3, . . .} be a sequence of scalars such that

then for every f ,

Proof.

Let f be an element of P . By expanding f in a Taylor series about

A,,,we find that

If p is an arbitrary positive number, then by multiplying and dividing the

182

Interpolation and Bases in Hilbert Space

[Ch. 4

summand by pk we find also that

Since P is closed under differentiation and I/ f ’ 1 5

rill f 11,

it follows that

By combining (2) and (3), we obtain

since

IA, - pnl 5 L . The result follows by choosing p

=

f i L . 1

Remark. It should be noted that the lemma applies whenever { A n } is separated and {ImA,} is bounded, and hence whenever {A,} is an interpolating sequence.

The next lemma provides a convenient criterion for insuring that a bounded linear transformation maps one Banach space onto another.

Lemma 4. Let T be a bounded linear transformation of the Banach space X into the Banach space Y. Suppose that there exist positive constants M and E , 0 < F. < 1, with the following property: for each vector y in the unit ball of’ Y there is a vector x E X such that

Then T maps X onto Y . ProoJ First observe that if Y E Y and llyli 5 A, then there is a vector E X such that IIxII MA and 1 ) T x - y 11 5 d.Let y be an arbitrary element in the unit ball of Y. It is clearly sufficient to show that y = Tx for some x . Begin by choosing x1 such that 1 x1 I/ 5 M and IIy - T x , 1 5 E . Next,

x

Sec. s]

Interpolation in the Paley-Wiener Space: Stability

183

with y - T x , in place of y and E in place of I , choose x2 such that 1 x2 1 5 ME and 11 ( y - T x , ) - T x , 1 5 E'. Proceed by induction. The result is a sequence lI ~ l x2, r x 3 , . . .) of vectors of X such that

Since 0 < E < 1, we have 1 1 1x, )I 5 C ME"- = M/(1 - E ) . Thus the series C x , is absolutely convergent, and so there is an element x in X to which the partial sums of the series converge. Since E, 0, (4) shows that y = T x . I --+

We are now in a position to prove that interpolating sequences in the Paley-Wiener space are stable.

Theorem 11. Let { 1 , , 1 21, 3 , .. .} be an interpolating sequencefor P. There exists a positive constant L with the property that {p,,p 2 , p 3 , . . .} is also an interpolating sequence for P whenever )I, - pnl iL for every n . Proof. Let p,,p 2 , p 3 , . . . be complex scalars such that [I,,- p,,l L for every n; the value of L will be specified later. Let T be the induced linear transformation of P into the vector space of all sequences defined by

The assertion is that if L is sufficiently small then TP = 12. Since {I,} is an interpolating sequence, ~ l f ( I , , ) 1S2 Bllfll' for some constant B and every ~ E Pand , hence the inequality

of Lemma 3 is valid. This shows that T is a bounded linear transformation from P into 1'. By Proposition 2 there is a constant M such that the system of equations

!(I,)

=

c,

(n = 1 , 2 , 3 , . . .)

has at least one solution f E P satisfying

provided that { c,}

E 1'.

(6)

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Interpolation and Bases in Hilbert Space

[Ch.4

The proof of the theorem is at hand. Let c = { c,} be an arbitrary element in the unit ball of l z . I f f is chosen in accordance with (6) and (7), then ( 5 ) becomes

By choosing L sufficiently small, we can make the right side less than 1, and the assertion TP = Iz follows from Lemma 4. I

Problems 1.

Let {Al, A’, A,, . . .} be a sequence of points that lie in a strip parallel to the real axis and let T be the induced linear transformation from P into the vector space of all sequences defined by

Show that if TP 3 P , then {A,} is separated, and hence TP c 1’. 2. Exhibit a bounded linear operator T on 1’ with the following two properties : (1) There is a constant M such that whenever Y E 1’ and ( ( y ( (5 1, we can find an element x E 1’ with 1 x 11 5 M and ( 1 T x - yll < 1. (2) T is not onto.

7

The Theory of Frames

We have seen that the study of Riesz bases in a separable Hilbert space H can be reduced to the study of an abstract moment problem:

In this section we shall introduce the notion of a “frame”, which will further elucidate the problem of moments and at the same time will provide yet another characterization of Riesz bases.

Definition. A sequence of distinct vectors { f l ,f z , f , , . . .} belonging to a separable Hilbert space H is said to be a frame fi there exist positive constants

Sec. 71

185

The Theory of Frames

A and B such that

for every j’ E H . The numbers A and B are called bounds for the frame.

It is clear that a frame is a complete set of vectors since the relations

(f, .A2) = 0 , n = 1, 2 , 3 , . . ., imply that f = 0. Notice also that the second inequality in (1) is just the condition that { ,fn} be a Bessel sequence with bound B. In practice, it is the first inequality that is difficult to verify.

Example 1. If { f,} is a complete orthonormal sequence in H , then ParseVal’s identity shows that (1) holds with A = B = 1. Example 2 (Nonharmonic Fourier Frames). A system of complex exponentials { e i A n ris} a frame in L 2 [ - n , n] provided that

I &) I 2

dt

for every 4 € L 2 [-n,n]. By virtue of the Paley-Wiener theorem, this is equivalent to

for every function f ( z ) belonging to the Paley- Wiener space. We associate with a given frame operator T on H defined by

{ , f l ,,f2, f 3 , .

. .}

a bounded linear

( T is a bounded operator because { j,} is a Bessel sequence: B C I ( f , f f l ) I 2 B211j’112).

s

Assertion: T is invertible. Since

n= 1

1 TfI12 5

Interpolation and Bases in Hilbert Space

186

[Ch. 4

it follows that

and so

This shows that T is bounded from below, and so must be one-to-one. But T is self-adjoint (indeed, positive), and an operator whose adjoint is bounded from below must be onto (see Taylor [1958 p. 2341). Thus T is invertible. Throughout this section T will denote the transformation defined by (2). Definition. A frame that ceases to be a frame when any one of its elements is removed is said to be an exact frame.

We are going to show that the class of Riesz bases and the class of exact frames are identical. It is convenient to first establish the following two lemmas. Lemma 5. Let { f l , f,, f 3 , . . .} be a frame in a separable Hilbert space H and f an arbitrary element of H . Then there is a unique moment sequence { a l ,a,, a3,. . .} such that m

f

=

c anfn,

n= I

and a,, = (9, fn),

If { b l ,b,, b 3 , .. .}

is

where

Tg = f .

any other sequence such that f =

I."=b,

fn,

then

Proof. The first statement is immediate since T is invertible and f T ( T - ' f ) . Suppose now that m

c anfn

n= 1

m

=

1 bnfn.

n= 1

=

Sec.71

187

The Theory of Frames

Taking the inner product of both sides with g, we find

n= 1

n= 1

and (3) follows readily. I

Lemma 6 . The removal of a vector from a frame leaves either a frame or an incomplete set. h o o f . Let { f n } be a frame in H,with bounds A and B , and suppose that f , is removed. By Lemma 5

where a, = (g,, f n ) and T g , = f,. We consider the two cases a, = I and a, # 1, showing that in the first case the system { f,},+, is incomplete, while in the second case it remains a frame. If a, = 1, then

and so

where a: = a, when n # m and a; b, equal to zero, we find

= 0.

Applying Lemma 5 , with every

n= 1

and hence a,, = 0 for n # m. Thus g , is orthogonal to f, for n # m. Since (g, f,) = a, = 1 , it follows that g m # 0 and hence that { f,},+, is incomplete. Suppose now that N, # 1. Then

Interpolation and Bases in Hilbert Space

188

[Ch. 4

Consequently, for all f in H

and so

where C that

=

1

+ 11

-

aml-’ CnZm lanl’. Since { f,,} is a frame, we conclude

n+m

where A , = A/C and B, = B . Thus the system { f n } n + m is a frame. I Remark. An examination of the proof reveals that if { f , , } is an exact frame and gn = T-lf,,, then {f,}and {a,} are biorthogonal.

We are now in a position to give the following characterization of Riesz bases.

Theorem 12. A sequence of vectors belonging to a separable Hilbert space H is a Riesz basis if and only if it is an exact frame. Proof. Let { f , , ] be a Riesz basis for H . Then it is complete and has I’ as its moment space. Accordingly, the mapping

defines a one-to-one linear transformation of H onto 1’. Since it is bounded (by Proposition 2), its inverse is also bounded (by the open mapping theorem), and the frame condition follows at once. Since the removal of a vector from a basis leaves an incomplete set, {j,,}is an exact frame. Suppose now that { f , } is an exact frame in H , with bounds A and B, and let f be an arbitrary element of H . Then by Lemma 5

where a,, = (9, f , ) and g

=

T - ’ f . To prove that this representation is

Sec. 71

The Theory of Frames

189

unique, suppose that

If we define yn by setting gn = T - ' f , , then { f n ) and {g,} are biorthogonal (by the Rrmurk following Lemma 6), and hence

since T is self-adjoint. Thus { j n is } a basis for H . It remains only to show that { f,} is a Riesz basis. By Theorem 1.7 this will follow if we can show that the series C c , f n is convergent if and only if C ( c n 1 2< a. Let the series c c , , f , be convergent, say to an element f of H . Then cfI= (g, ,f,),where y = T ' Jand , so by the frame condition

Suppose now that {c,} is any sequence of scalars such that cIc,,12< 00. Then for n 2 m we have by Theorem 3

since { . f n } is a Bessel sequence with bound B. Since the right side tends to zero as m,n + m, the partial sums of the series cnfn form a Cauchy sequence, and hence the series is convergent. I

Problems 1.

Let ( f ' l , 1;. f 3 , . . .} be a frame in a separable Hilbert space H and put gn = T - ' J , . Prove that {y1,y2,y3,.. .} is also a frame and that for

every f , c€

f' 2.

=

Prove that if { , f l , f 2 , then

c

n= 1

f3,.

ACIC"12

whenever { c,,} E 1'.

c UI

(f?clfl)jn

=

(f,fn)sn.

n= 1

. .} is an exacf ,frame, with bounds A and B,

s IICcnf,1I2 IBCICn12

190

Interpolation and Bases in Hilbert Space

[Ch. 4

3. (DuffinSchaeffer) Let { f l , f 2 , f3,. . .} be a frame, with bounds A and B , and let p = 2/(A + B).If h is an arbitrary vector, we define a sequence h“), h“), h(2),. . . recursively by setting h ( O ) = h and h ( k + 1)

=

-p

f

fn)f,

for k = 0,1,2,.. ..

n= 1

Let

and

n= 1

Prove that

(Hint: If S

=

I - p T , then

8 The Stability of Nonharmonic Fourier Series In this section we shall discuss the stability of the class of Riesz bases

{eiLn‘}in L 2 [ -n, XI, first under “small” displacements of the ,In’s and then

under more general “vertical” displacements. As a result, we shall find that Kadec’s *-theorem can be dramatically improved.

Sec. 8l

The Stability of Nonharmonic Fourier Series

191

The modus operandi is to combine the stability of interpolating sequences with the stability of frames.

Theorem 13. l j the system { eian'}is a frame in L2[ - n,n], then there is a positive constant L with the property that { eipn') i s also a frame in L2[ - n,n] whenever ]A,,- pnl 5 L .for every n. Proof. Let {eiAnr} be a frame in L 2 [ -n, n], with lower and upper bounds A and B , respectively. Then

for every function f belonging to the Paley-Wiener space P. Let L be a positive number and p,,p 2 , p 3 , . . . complex scalars for which 11, - pnl 5 L ( n = 1,2, 3 , . . .). It is to be shown that if L is sufficiently small, then similar inequalities hold for the pn's. By virtue of Lemma 3, for every j in P,

where C

=

B -(enL A

-

Applying Minkowski's inequality, we find that

and hence that

for every j . Since C is less than 1 if L is sufficiently small, the system {eipnr} is a frame in L 2 [ -71, n].

Corollary. If the system ( e i a n fis) a Riesz basis for L2[-n, 113, then there is a positive constunt L with the property that {eipnr} is also a Riesz basis for L 2 [ -n, 713 whenever [A,,- pnl L for every n.

192

Interpolation and Bases in Hilbert Space

[Ch. 4

Proof. Let {eianf} be a Riesz basis for L z [-71, n] and suppose that pl,p,, p , , . . . are complex scalars for which sup,lA, - pnl = L < 00. If L is sufficiently small, then (1) {p,} is an interpolating sequence for P (by Theorem 11) and (2) { e i M nisf } a frame in L z [- A , n] (by Theorem 13) and hence complete. Now invoke Theorem 9. I The criterion of Theorem 13 is far too restrictive-the essential ingredient is not that [A,,- p,I is small but rather that Re@, - p,) is small. More precisely, we have the following result (cf. Theorem 3.12). Theorem 14. Let { A l , A,, A,, . . .} be a sequence of points lying in a strip parallel to the real axis. If the system {ei(Re a n ) 1 } is a frame in L z [ -n, n ] , then so is { e i L f } .

+

Proof. Write A,, = u, ip,, with u, and p,, real, and choose M so large that lPnl 5 M for every n. Suppose that the system {eianr}is a frame in L2 [-IT, n], with lower and upper bounds A and B , respectively. Then

for every function f E P. We shall establish similar inequalities for the An’s. One half is easy. Since [A, - a,[ 5 M , it follows from Lemma 3 that

for every f . A straightforward application of Minkowski’s inequality then shows that

This establishes one half of the frame condition. It is convenient to prove the other half in three steps.

(i) Let f ( z ) be a nontrivial function in P. By Hadamard’s factorization theorem

Sec. 8 1

193

The Stability of Nonharmonic Fourier Series

where z l , z 2 , z 3 , . . . are the zeros of . f ( z ) other than z = 0, k is the order of the zero at the origin, and a and b are complex numbers. Since .f'(z)is bounded on the real axis, the series

, ~a, is absolutely convergent by Theorem 2.14. If l / ~ = where a,, and h,, are real, then

+ ib,,,

where c = Re(a) and d = Im(a) + b,. Since t,,e product in (1) converges uniformly on each compact set, so does the product in ( 2 ) . (ii) There exists a sequence A\'), A;'), I.\'', . . . of complex scalars and an entire function f1(z) belonging to P such that Rei),"

=

Rei,,

\lrni!;)\ 5

M 2

(n = 1,2,3,. . .)

(3)

and

Assume that d 2 0. (The case d < 0 is similar and is left to the reader.) We begin by defining a function g(z) whose zeros are obtained by reflecting across the real axis those zeros of f ( z ) that lie in the lower half-plane. Specifically, we set z,~

z,

and

if Imz, 2 0, if Imz, < 0,

Interpolation and Bases in Hilbert Space

194

[Ch. 4

Assertion: g(z) E P and whenever x is real.

Ig(x)l = If(x)I

(6)

To prove the assertion, define a sequence gl(z), g2(z),g3(z),. . . of entire functions by writing

n 1m

gm(z) = f ( z )

- z/w,

1

z/z,

,

m = 1,2,3,....

It is clear that g,(z) + g(z) as m+ co, uniformly on every compact set, and hence g(z) is entire. It is also clear that g,(z) and f(z) have the same modulus on the real axis, and this establishes (6). It remains only to show that g(z) is of exponential type at most II. For each positive integer m the quotient g,(z)/f(z) approaches 1 as IzI + 00, and hence

for all z. Fixing z and letting m + 00, we see that g(z) is an entire function of exponential type at most n. This proves the assertion. Observe now that Ig(z)l

5 If(z)l

whenever

Imz 2 0

(compare formulas (2) and (5)) and Ig(z)( 5 If(Z)l

whenever I m z 2 0

since d 2 0. It follows that if

then

Sec. s]

The Stability of Nonharmonic Fourier Series

195

Finally, define

Then (3) holds and lfl(L!,'))I 5 (f(A,,)l for every n. Thus, to establish (4), we have only to show that

Since g(z) E P, there is a function

4 E L 2 [-n, n] such that

Since f ( z ) and y(z) have the same modulus on the real axis, we have

by the Plancherel theorem. If we now set & ( t ) = r$(t)e-M"2,then f l ( z )=

and hence

{;a,

If'l(X)12 d x =

-

Lr

2n -"

r

$,(t)eiz' dt,

m

14(t)12e-M'dt2 eCnM

2n ~"

If(x)12d x .

Thus IIf,112 2 e-nMIlf112. (iii) We have completed the first step of an obvious induction procedure. Accordingly, given f , we can find for each positive integer k a sequence A i k ) , A f ) , A f ) , . . . of scalars and an entire function fk(z) belonging to P such that Re Ahk) = Re A,,,

IIm Akk)l 5

M 2k

-

(n

=

1 , 2 , 3 , . . .)

Interpolation and Bases in Hilbett Space

196

[Ch. 4

and

+

+

* . . (The constant 271M has been written in place of n M i n M ( i ) k - l n M . ) Since the set {eian'}is a frame, there is a positive constant L such that {eiyn'}is also a frame whenever la,,- ynl 5 L by Theorem 13. Choose k so large that M / 2 k 5 L. Then the set {eiAF)'}is a frame, and the left side of (7) has a positive lower bound A , , independent off. Hence

+

for every f EP. I Corollary 1. Let {Al,A 2 , 1 3 , .. .} be a sequence of points lying in a strip parallel to the real axis. I f the system { ei(Re is a Riesz basis for L 2 [-71, n], then so is {e'"'}.

Proof. It is sufficient to prove that the system {eiAn'}is an exact frame in L 2 [ -n, n]. That it is a frame follows from Theorem 14 since In)'} is a frame. Suppose then that a single arbitrary term eiAk'is deleted. Assertion: {ei(Rean)'}nfk is incomplete in L 2 [ - n , n]. Reason: the removal of a vector from a frame leaves either a frame or an incomplete set. Concluis incomplete in L z [ -n, 711 by virtue of Theorem 3.12 and sion: {eiAn'},,zk so cannot possibly be a frame. Thus the system {einn'}is an exact frame. I The following impressive generalization of Kadec's immediate. Corollary 2.

a- theorem is now

I f {A,,}?, is a sequence o j scalars f i r which supIRei, - n] < n

and

suplImA,I < co, n

then the system { eiAn'}Tm is a Riesz basis .for L 2 [--n, x]

Every example of a Schauder basis {eiLn'}for L 2 [ -n, 713 seen so far was proved to be a Riesz basis. In a sense this is not surprising-the class of

Sec. 9 1

Pointwise Convergence

197

Riesz bases is very large. Question: Are there bases of complex exponentials that are not Riesz bases? An example has not yet been found.

Problems 1.

2.

How must the proof of Theorem 14 be modified when d < O? Let { e i A n fbe } a basis for L 2 [ -n, n] and suppose that supnIIm1,1 < Prove that { e i P n fis) also a basis for L 2 [ - n,n] provided that

00.

( H i n t : By virtue of Theorem 1.12, it is sufficient to prove that

n= 1

9

n= 1

Pointwise Convergence

The time has come to consider pointwise convergence. With regard to ordinary Fourier series, the deepest and most striking result is surely Carleson’s proof of Lusin’s conjecture: the Fourier series of every L2 function converges pointwise almost everywhere. The following general “equiconvergence” theorem will show that nonharmonic Fourier series in L2 have to a large extent the same convergence and summability properties as ordinary Fourier series. Recall to begin with that two series x u n and C b , are said to be equiconvergent if their difference x ( a n - b,) converges and has the sum 0. Theorem 15. Let {e‘”‘)?,, be a Riesz basis for L 2 [ - x , n ] and suppose that

supl1, - n( <

GO.

n

For each function in L z [ -n, n] the ordinary Fourier series and the nonharmonic Fourier series are uniformly equiconvergent on every compact subset qf ( -n, n).

198

Interpolation and Bases in Hilbert Space

[Ch. 4

Proof. Let f be an arbitrary element of L 2 [-n, n] and 6 a small positive number. It is to be shown that i f f has the two norm-convergent expanthen the difference sions aneintand cneiAnt,

c

1

N

C

(a,e""

-

cneiAnt)

n=-N

+

converges to zero as N + co, uniformly on the interval [ --71 6, n - 61. The essence of the proof is to obtain a suitable representation for this difference. Begin by writing

where ik(& - n)k bnk

=

k!

( k Z O , - c o < n < co).

If f N denotes the Nth partial sum of the nonharmonic Fourier series of f and

then

Choose L so large that 11, - nl 5 L for every n . Then

1 $Nk(I2

N

=

1

Ic,bnk12

n=-N

r 2k

and hence

N

(by Parseval's identity)

Sec. 9]

199

Pointwise Convergence

exists for every k . (Here 1.i.m. means limit in mean in the L2 sense.) Observe that = c < x. and 1 5 cLk/k! for k 2 0. Assertion :

( ., I2

Indeed, since 1 1 1t k $ k ( f ) I ( 5 ~ x ( n L ) ~
and since lim,II $k - $ N k ( I = 0 for each k , we must have lim,II g -I(,/ = 0. But lim, 11 f - f , 1 = 0, and therefore f ( t ) = g ( t ) almost everywhere on [ --n, x]. This proves the assertion. Let D, be the Dirichlet kernel?, i.e., DN(4 =

+

sin(N +)t sin +r

( N = 1,2,3,. . .),

and g, the Nth partial sum of the ordinary Fourier series o f f . Then

It follows at once from the definitions of

$Nk

and

t To simplify the notation, we have suppressed a factor of:

$k

that

200

interpolation and Bases in Hilbert Space

[Ch. 4

and hence N

C

m

(a,,e'"' - c,,eiAn')=

n= -N

($k(X),

(xk - c ~ ) D ~-( tx) )

k=O

whenever It1 2 n. This is the desired representation. We complete the proof by estimating the inner products

Denote by Q the closed rectangle {(x, t ) : l x l the function Hk(x,f) = .

Xk

- tk

for ( x , t ) E R

sint(x - t )

n, It1 5 n - 6 } , and by Hk

(k = 0, 1,2,. . .).

It is clear that each H , is continuous on R. It is not difficult to show (the proof is left to the reader) that there is a constant A such that

Therefore, if It[ 5 n - 6 , then

for all values of N and k . Given

E

> 0, choose M so large that

c -(nL)k < -. k! CA E

k>M

Then, for every N ,

Since Hk is uniformly continuous on Q, a simple extension of the RiemannLebesgue lemma shows that

Sec. 91

201

Pointwise Convergence

uniformly for (tl 6 n - 6. This proves that

uniformly on each compact subset of

(-71,

n). I

We conclude that the nonharmonic Fourier series of an L2 function converges at a given point of ( - n , n ) if and only if the ordinary Fourier series converges at that point; by Carleson's theorem, this occurs almost everywhere. Furthermore, on every compact subset of ( - n , n), the summability properties of the two series are uniformly the same (Problem 2).

Problems 1.

Let Hk and R be defined as in Theorem 15. Show that there is a constant A such that

2.

Prove that Theorem 15 remains valid when the term "equiconvergent" is replaced by "equisummable", for any summability method that is both regular and linear. (For an excellent treatment of such methods, which include (C, l), Abel, and Riesz summability, see Szasz [1944].) 3. Let be a sequence of complex numbers for which supIReII, - n ) < n

and

suplIrnA,I < co. n

Prove that the system {eiaf"}3"1,is complete in C(Z), for each closed subinterval I of ( - n , n). 4.

By modifying the proof of Theorem 15, show that N

(n - It])

1 (a,,e'"' - cneiAn') + 0 -N

uniformly on [ - n, n].

as N + m ,

This Page Intentionally Left Blank

NOTES AND COMMENTS

CHAPTER 1 Section 1 General references. The first systematic investigation of bases in Banach spaces was made by Banach [ 19321. Since then, and especially over the past two decades, research in this area has flourished. Two excellent references, the first an introductory text, the second an encyclopedic account of bases in Banach spaces, are by Marti [1969] and Singer [1970]. The important work of Lindenstrauss and Tzafriri [1977] deals mainly with recent results and current research directions. Weak bases. The definition of a basis for a Banach space can be weakened. The "weak basis theorem" asserts that a weak basis for a Banach space, i.e., a basis relative to the weak topology, is in fact a Schauder basis. This is stated without proof in Banach [1932, p. 2381. Karlin [1948] sketches a proof. A complete proof, valid for every Frechet space, is given in Bessaga and Pelczynski [ 19591. Bases for special spaces. Most of the common Banach spaces are now known to possess bases. Among the many important examples, the following deserve special mention. In L P [ - n , n], 1 < p < co, the trigonometric system {ei"')? and in LP[O, 11, 1 5 p < 00, the H a m system { h , } constitute Schauder bases. By definition, h,(t) = 1 and

hzk+,(t) =

1

- 1,

0,

t E [(2m - 1)/2'+ ', 2m/2'+

elsewhere 203

'1

Notes and Comments

204

( k = 0, 1,2,. . . , m

=1

and

=

1,. . ., 2k)(see Marti [1969, p. 491). If we put =

J

h,(u) du

for 0 5 t 5 1 ( n = 1,2,3,. . .),

0

then we obtain the Schauder system {4,,}; it is a basis for C[O, 11 (see Section 2). More recently, Billard [1972] has constructed a basis for the Hardy space H' ; Ciesielski and Domsta [ 19723 and Schonefeld [19721 have independently constructed a basis for C k ( P ) ;and BoEkarev [ 19741 has constructed a basis for the disk algebra A , using the Franklin system in an essential way. (The Franklin system is obtained from the Schauder system by applying the Gram-Schmidt orthogonalization procedure. It is also a basis for C[O, I]. For a detailed study of the Franklin system, see Ciesielski [1963, 19661.) For further examples see Pelczynski [ 19771.

Problem 8. It was shown by Riemann that in a finite-dimensional Banach space the class of absolutely convergent series and the class of unconditionally convergent series coincide. In an infinite-dimensional Banach space, they do not. Theorem (Dvoretzky and Rogers [1950]). If X is an infinite-dimensional Banach space and if { c,} is an arbitrary square-summuble sequence of positive numbers, then there are elements x, E X such that 1 x, 1 = c, (n = 1,2,3, . . .) and the series x, is unconditionally convergent. Conditional and unconditional bases. A basis {x,} for a Banach space is said to be unconditional if every convergent series of the form c,x, is unconditionally convergent. The trigonometric system is unconditional in Lp[-n, n] only when p = 2 (Karlin [1948]). The following two systems are both unconditional in Lp[O,11 for 1 < p < W : the Haar system (Marcinkiewicz [19371) and the Franklin system (Ciesielski, Simon, and Sjolin [ 19771). Since Lp[O,11 and H p are isomorphic for 1 < p < m (Boas [ 1955]), it follows that lP (1 < p < m) also has an unconditional basis. It is not yet known whether HI does. (For an excellent account of HP-spaces, see Hoffman [1962] and Duren [1970].) There are also spaces that admit no unconditional bases: C[O, 11 (Karlin [1948]) and L'[O, 13 (Pelczynski [1961]), for example (cf. Arutjunjan [1972].) In this connection the following result due to Krotov [1974] is striking: There exists a continuous function on [0,1] whose expansion in the Schauder system, with a suitable ordering of the terms, converges almost everywhere to any prescribed measurable function. The first example of a reflexive separable Banach space that admits no unconditional basis was given by Kwapien and Pelczynski [1970]

IF=

205

Chapter 1

(see also Lindenstrauss and Pelczynski [1971]). Since there are spaces in which all bases are conditional, it is natural to ask if there are spaces in which all bases are unconditional. The answer is no-every Banach space with a basis has a conditional basis (Pelczynski and Singer [1964]; see the Notes for Section 8). The approximation property. A Banach space X is said to have the approximation property (in the sense of Grothendieck [ 19553) if the identity operator on X can be approximated, uniformly on every compact subset of X , by operators of finite rank. Enflo's example provides a separable Banach space that fails to have the approximation property and so a fortiori fails to have a basis (see Problem 7, Section 6). The example has been greatly simplified by Davie [1973], who shows that the spaces c, and lp ( 2 < p < co)have subspaces that fail to have the approximation property. It is worth noting that it is usually much easier to verify that a given space has the approximation property than to show that it has a basis. Thus, for example, it is relatively simple to show that the disk algebra A has the approximation property (see Problem 8, Section 6), while the existence of a basis for A was for many years an open problem.

Section 2 The Weierstrass approximation theorem and related theorems. The classical theorem of Weierstrass on the approximation of continuous functions by polynomials appears in Weierstrass [ 18851. Bernstein's proof, which is based on probability theory, is from Bernstein [1912a]. (For a penetrating study of the Bernstein polynomials, see Lorentz [19531.) Weierstrass's theorem has been generalized in many different directions. The important extensions of Stone, Muntz, and Mergelyan are well known (see, e.g., Rudin [1966, pp. 305, 3861 and Royden [1968, p. 1741). We mention, in addition, the following result of Wolibner [1951] on simultaneous interpolation and approximation: [f'f E C[a, b] and if {xl, . . .,x,} is a set 91' n arbitrarjq points in the interval [a, b ] , then for each E > 0 there is a poll'nomiul p such that (1) 11 f - p 1 ) < E , ( 2 ) lip 1 = 1 f 1 , and (3) p(x,) = f ( x i ) . i = 1,. . . ,n. (This generalizes a well-known result of Walsh [1935, p. 3101 in which properties (1) and (3) were shown to hold.) In Deutsch and Morris [ 19691, property SAIN (simultaneous approximation and interpolation which is norm-preserving) is introduced and studied in an abstract normed vector space.

206

Notes and Comments

Theorem 1. Schauder [1927]. Meletidi [1972] has shown that for arbitrary positive numbers A and B ( A < B) there exists a basis { f n } for C[O, 11 such that A 5 f,(t) 5 B for n = 1, 2, 3, . . . and all values of t E [0, 11.

Section 3 General references. For Hilbert space theory, Halmos [1951] and Berberian [1961] are elementary references; for a more advanced treatment, see Riesz and Nagy [1955], Dunford and Schwartz [1958, 19631, and Halmos [1967]. For HP-spaces, see Hoffman [1962] and Duren [1970]. And for Fourier series, Zygmund [1959] is the standard work; for a more elementary introduction, see Titchmarsh [19391. Problem 6. This definition of orthogonality was introduced by Birkhoff [1935] and extensively developed by James [1945]. Problem 8. The theory of almost periodic functions, an elegant and natural generalization of the theory of periodic functions, was created by Bohr [1925a, 1925b, 19261. (For an excellent account of the elementary theory, see Bohr [1947].) Since then it has been considerably extended (see, e.g., Stepanoff [ 19251, Besicovitch [ 19321, and Paley and Wiener [1934, p. 1161; other references may be found in Dunford and Schwartz [1958]). Problem 12. Vitali [ 19211. Problem 13. Dalzell [ 19451. Problem 14. For an excellent account of the classical orthogonal polynomials, see Szego [1939]. Problem 16. These facts were first discussed by Bergman [1947, p. 241. The suggested proof is from Halmos [1967, p. 1871.

Section 4 General references. Aronszajn [ 19501 and Hille [19721. Additional references may be found in Shapiro [1971, p. 1011. For a deeper discussion

207

Chapter 1

of the Bergman kernel and its application to conformal mappings, see Bergman [ 19501 and Nehari [ 19521.

Section 5 Problem 6. Boas and Pollard [ 19481. Problem 8. A classical treatise on closure theorems for analytic functions is Walsh [ 19351 (see also Davis [ 19631).

Section 6 Theorem 3. Banach [1932, p. 1111. Schauder's original definition of a basis had required the continuity of the coefficient functionals. Theorem 4. James [ 19501. Problem 5. The term "normal basis" was introduced by Karlin [1948], although the concept already appears in Banach [1932, p. 2381. The suggested proof is from Diestel [ 1975, p. 431. For another simple proof, see Cook [1970]. Problem 7. For a detailed investigation of the many variants of the approximation property and the relations between them, see Grothendieck [1955].

Section 7 Bases for the dual space. Let X be a Banach space with a basis. Even if XY is separable, it need not have a basis. This is hard-the proof makes use of the existence of a separable Banach space without a basis (see Lindenstrauss and Tzafriri [1977, p. 341). A deep result of Johnson, Rosenthal, and Zippin [I9711 states that if F' has a basis, then so does X . (The problem had been posed by Karlin [19481.) Theorem 5. This result is essentially due to Banach [1932, p. 1071. Problem 3. Karlin [1948].

Notes and Comments

208

Section 8 Equivalent bases. Equivalent bases were studied by Arsove [I9581 (he calls them “similar bases”). Theorem 7 was proved by Arsove [I9581 for locally convex Frechet spaces and, independently, by Bessaga and Pelczynski [19581. It has recently been shown by Ciesielski, Simon, and Sjolin [I9771 that the Haar and Franklin systems are equivalent in Lp[O,11, 1 < p < m. Sjolin [1977] has shown that this result is not true i f p = 1. Uniqueness of bases. Once a Banach space is known to have a basis, it is natural to ask about its “uniqueness”, up to equivalence. In a finite-dimensional Banach space all bases are equivalent. In an infinite-dimensional Banach space the situation is dramatically different-bases, if they exist at all, are never unique. Theorem (Pelczynski and Singer [19641). In every injnite-dimensional Banuch space with a basis, there exist uncountably many nonequivalent normalized conditional bases. (Normalized means that each basis element has norm 1.)

If a basis for a Hilbert space is to be equivalent to an orthonormal basis, i.e., if it is to be a Riesz basis, then it must be at the very least both bounded and unconditional. Remarkably, the converse is also true: in a separable Hilbert space all bounded unconditional bases are equivalent. This observation goes back to Kothe [I9361 and Lorch [1939]. It was rediscovered by Gelbaum [1950] and Gelfand [1951]. Since then it has been reproved many times (see the references in Singer [1970, p. 6401). The quickest proof (see Lindenstrauss and Tzafriri [1977, p. 711) makes use of Orlicz’s theorem: I f c x n is an unconditionally convergent series in a Hilbert space, then cIIxnl12 < cu (Orlicz [1933]; for a one-line proof of Orlicz’s theorem, see Lindenstrauss and Tzafriri [1977, p. 181). The spaces co and I’ also have the property that all bounded unconditional bases are equivalent (Lindenstrauss and Pelczynski [ 19681). There are no other examples. The only infinite-dimensional Banach spaces with an unconditional basis, in which all normalized unconditional bases are equivalent, are (up to an isomorphism) co, I‘, and l2 (Lindenstrauss and Zippin [ 19691).

Theorem 9. Bari [ 19511. Conditional bases for Hilbert space. The first example of a conditional basis for L 2 [ --n, n] was given by Babenko [I9481 (see also Gelbaum

209

Chapter 1

[I9511 and Bari [1951]). For a simple example of a conditional basis for 12, see Lin and Singer [1971]. Although we have dealt almost exclusively with unconditional bases, the following result deserves attention : Theorem (Gurarii and Gurarii [ 19711). [f [ , f n } is a normalized basis f o r a Hilbert space H, then there exist positive constants A , B, p , and q, with 1 < p 5 2 5 y < x,such that for each convergent expansion .f = c, f , ,ce haw

(For a different proof, see Ruckle [ 19721.)

Section 9 General references. Singer [ 19701 and Retherford and Holub [19711. Theorem 10. The original form of this theorem, with X = L*[a, b] and [ x n l an orthonormal basis for X , is due to Paley and Wiener [1934, p. 1001. The generalization given here is due to Boas [1940]. For further results in this direction, see Singer [1970, pp. 84-109, 337-3591. Generalizations in a different direction are given by Bourgin [ 19461, Gosselin and Neuwirth [ 1968/69], and Davis [ 19691. Corollary (Theorem 10) and Theorem 12. Krein and Liusternik [1948]. Theorem 11. Krein, Milman, and Rutman [ 19401. An extension to locally convex spaces has been given by Lerer [1969]. The following result of Gurarii and Meletidi [I9701 had been for many years an open problem. Theorem. tf (slI> is a complete sequence of elements ,from a Banach space X , then there exist positive numbers E, such that any sequence {y,} for which IIx, - 1 < ell (n = 1, 2, 3, . . .) is also complete in X . Problem 3.

Retherford and Holub [ 19711.

Section 10 Nonharmonic Fourier series. The study of nonharmonic Fourier series was initiated by R. E. A. C . Paley and N. Wiener in their celebrated treatise of 1934. In this work the criterion of Theorem 13 is introduced and verified

Notes and Comments

210

whenever each An is real and IA,,- nl L < l/n2 (n = 0, f 1, f 2 , . . .). Ultimately, it was shown that the constant l/n2 could be replaced by 4 (Kadec [1964]; see Theorem 14). That is in fact the "best possible" constant follows from a theorem of Levinson [1940, p. 481 : The system of exponentids (efi("-*)':n= 1,2,3,. . .} is complete in L z [ - n , n] (see Theorem 3.5). The theory of nonharmonic Fourier series has found practical applications in such diverse areas as the theory of diffusion (see Hammersley [1953]), control theory (see Russell [1967]), and the solution of the neutron slowing down equation (see Sengupta and Karnick [ 19781). Theorem 15. Bari [1951]. Theorem 16. Birkhoff and Rota [1960]. Sturm-Liouville series. The idea of expanding an arbitrary function in terms of the solutions of a second-order differential equation goes back to the time of C. Sturm and J. Liouville (see ,Titchmarsh [1946]). The proof of Theorem 17, based on Theorem 16, is due to Birkhoff and Rota [1960]. For a generalization of Theorem 16, together with an application to certain non-selfadjoint boundary value problems, see Brauer [19641 (see also Wallen [ 19691). Theorem 18 and Problems 1, 2. Duffin and Eachus [1942]. The suggested proof of Problem 2 is due to Riesz and Nagy [1955, p. 2091. Problem 3. Schafke [1949]. The suggested proof is from Retherford and Holub [ 19711. Problem 4. Birkhoff and Rota [1960]. A somewhat simpler argument may be found in Halmos [1967, Problem 71.

CHAPTER 2 General references. Bore1 [ 19211, Valiron [ 19491, Boas [ 19541, CartWright [ 19561, Levin [19641, Markushevich [19661, and Holland [ 19731. See also Titchmarsh [19391 and Markushevich [19651.

211

Chapter 2

Part One

Section 1 Information on infinite products in general can be found in Knopp [1951]. Theorem 1. Weierstrass [ 18761.

Section 2 Theorem 2. Jensen [ 18991. For alternative proofs, see Titchmarsh [1939].

Section 3 The various possible relations between p , Shah [1941].

A, and

p are illustrated in

Section 4 Theorem 6. Bore1 [1897] Theorem 8. For more precise results, see the Notes for Section 5.

Section 5 Theorem 9. Hadamard [18931. For another proof, see Titchmarsh [1939, p. 2501. The minimum modulus. Let f ( z ) be a nonconstant entire function and let m(r) be its minimum modulus for IzI = r. It follows readily from Hadamard’s factorization theorem that if f ( z ) is of order p , then for each E > 0,

on circles IzI

=

r of arbitrarily large radius (see Problem 4).Under special

212

Notes and Comments

conditions, much more can be said. For example, if 0 < p < I , then there are arbitrarily large values of r such that m(r) > (M(r)),osRp-s

(see Titchmarsh [1939, p. 2751). For further results, see Boas [1954] and Holland [19731. Problem 11. See Titchmarsh [1939, p. 2661.

Part Two

Section 1 Theorem 10. Phragmen and Lindelof [1908]. For additional theorems of this type, see Titchmarsh [19391. Theorem 11. See, e.g., Polya and Szego [1925, Part IV, Problem 2021, Duffin and Schaeffer [1938], and Plancherel and Polya [1938]. If ,f’(z) is real for real z, then there is a sharper result : If(x

+ iy)l 5 M cosh By;

moreover, if equality holds for a single nonreal point, then f ( z ) = M cos(Bz C), where C is real (Duffin and Schaeffer [1938]; for another proof, see Redheffer [19531). It is frequently possible to determine the growth of an entire function of exponential type along a line from its growth along a sequence of points on or near the line. For example, Cartwright [1935] has shown that an entire function of exponential type < I Tthat is bounded on the integers must be bounded on the entire real axis. This was generalized by Duffin and Schaeffer [1945], who showed that the integers can be replaced by any sequence having un$orm density 1. (A sequence { l , } ? ,of real or complex numbers is said to have uniform density 1 if it is separated and sup, [I,- n( < GO.) There are similar results when { f ( l , ) } E l p (see the Notes for Section 3).

+

Bernstein’s inequality. Bernstein proved the inequality named after him first for trigonometric polynomials (Bernstein [ 1912bl) and then later for arbitrary functions in B, (Bernstein [19231). The following generalization

213

Chapter 2

is due to D u f i and Schaeffer [1938] (see also Akhiezer [1956, p. 1441): If f ( z ) E B, and is real for real z, then

This shows not only that ‘lf’(x)I 5 T supI,f(x)l but also that If’(x)l can be large only when If’(x)l is small. For further generalizations and improvements, see Boas [1954, pp. 210-2181 and Giroux, Rahman, and Schmeisser [ 19791. Problem 9. The suggested proof is due to Polya and Szego [1925, Part 111, Problems 295, 2981.

Problem 10. See Theorem 14. For yet another proof, see Luxemburg and Korevaar [1971, p. 261 (cf. Levinson [1940, p. 251). Problem 12. This proof is due to Boas [1954, p. 2101. Problem 13. For another simple and elegant proof, see O’Hara [1973].

Section 2 Theorem 13. Carleman [ 19221. Here “Carleman’s formula” is introduced for the purpose of proving Miintz’s theorem. Muntz’s theorem (Muntz [1914], Szasz [1916]). For a very good review of work on Muntz-Szasz type approximation since N. Bernstein proposed the problem in 1912, see Luxemburg [1976]. Of the many proofs of Miintz’s theorem that have been given, the following deserve attention: Polya and Szego [1925, Part IV, Problem 1981, Paley and Wiener [1934, p. 321, 1954, p. 2351, Akhiezer [1956, p. 43 , Rudin [1966, p. 3051, Feller , Luxemburg and Korevaar [1971? and Shapiro [1971, p. 991. The converse is also true: if l/An < co, then the system {tan) is incomplete in C[a, b]. In this case a complete characterization of the closed linear span of the system {tan}was provided by Clarkson and Erdos [1943], Korevaar [ 19471, and Schwartz [1959]. For an expository account of these and related results, see Anderson [1978].

1

Density. If f ( z ) is an entire function of exponential type T > 0, bounded on the real axis, then the zeros of f ( z ) in each half-plane x > 0 and x < 0

214

Notes and Comments

have density ~ / a(Levinson . [1940, Chap. 1111 proves an even stronger result.) The first result of this sort appears to be due to Titchmarsh [1926, p. 2851. Problem 4. See Titchmarsh [1939, p. 1321. Problem 5. Titchmarsh [1926, p. 284; 1939, p. 1371.

Section 3 All the results of this section are due to Plancherel and Polya [1938]. See Boas [1954, pp. 97- 1031 for a wide range of generalizations. Theorem 17. There is a variety of converse results when f ( z ) is of sufficiently small type. If, for example, f ( z ) is of exponential type T < a and if for some positive number p, I T a If(n)lP< 00, then

where B depends only on p and z (Plancherel and Polya [1938]; see also Boas [1940]). An analogous result is valid if the integers are replaced by any sequence having uniform density 1 (Duffin and Schaeffer [1952] for p = 2; Boas [1952]). Problem 2. See Boas [1954, p. 991. Additional references may be found in Boas [1938]. Problem 4. For a sharper estimate, see Boas [1954, p. 1021. Problem 7. Plancherel and P6lya [1938]. When p 2 1, A can be taken to be T~ (see, e.g., Boas [1954, p. 2111).

Section 4 Theorem 18. Paley and Wiener [1934, p. 121. The proof given in the text is from Boas [1954, p. 1051. For additional proofs, see Boas [1954, p. 1061, Rudin [1966, p. 3711, and de Branges [1968, p. 471. There are analogous Fourier representations for entire functions of exponential type

215

Chapter 3

that belong to LP on the real axis, as well as for functions that are analytic and of exponential type in a half-plane and belong to Lp on the boundary (see, e.g., Boas [1954, p. 1071 and Rudin [1966, p. 3681). Example 2. This proof is due to Boas [ 19371. Example 3. Boas [ 19721. Problem 3. This problem appears in Boas [1954, p. 2441.

Section 5 The Paley-Wiener space and other Hilbert spaces of entire functions are discussed from a different point of view in de Branges [ 19681. For an extensive treatment of the cardinal series, see Whittaker [1935]. Problem 3. See Boas [I9721 and Pollard and Shisha [ 19721. Problem 4. See, e.g., Young [1974a, p. 1121.

CHAPTER 3 General references. Paley and Wiener [ 1934, Chap. VI], Levinson [ 19401, Schwartz [ 19591, and especially Redheffer [ 19771.

Section 1 The terms exact, excess, and deficiency were introduced into mathematical terminology by Paley and Wiener [1934, p. 921. Corollary (Theorem 1). The following alternative proof, based on the F. and M. Riesz theorem, was suggested by P. S. Muhly. If p is a complex Bore1 measure on the interval [ --n, n] such that

n:J

einrd p ( t ) = 0

( n = 1, 2, 3,. . .),

216

Notes and Comments

then p is absolutely continuous with respect to Lebesgue measure and d p / d t c H ' . (As is customary, we have identified H' with the subspace of L1[ -n, n] spanned by the exponentials ei"' with n 2 0.) Since a function in H' cannot vanish on a set of positive measure unless it vanishes identically (see Hoffman [ 1962, p. 52]), the result follows.

Theorem 2. Let {Al, A,, ,I3,.. .} be an increasing sequence of positive numbers. Szhsz [1933], solving a problem of Pblya [1931], proved that if lim inf n/A, > 1, then the conditions f E C [ -n, n]

and

f ( t ) cos Aflt dt = 0

imply that f is identically zero. Under the weaker assumption lim sup n/ A,, > 1, Paley and Wiener [1934, p. 841 were able to show that if

.f E L 2 [-n, n]

and

1"

J

f(t)e*'"' dt

= 0,

n

=

1,2,3,.

. .,

--1[

then f ( t ) = 0 almost everywhere on [ -n, n], while Levinson [1935] derived the same conclusion when

(For an even stronger result, see Theorem 14.) Problem 4. This is essentially Theorem 1 in Levinson [1940].

Section 2 The substance of this section is taken from Levinson [1940, Chap. I, Section 31.

Problem 3. Levinson [ 1940, p. 61. Redheffer [ 1977, p. 341 has established the following sharp result for p = 2: If { A f l } is a sequence of real or complex numbers satisfying lAn - nl 5 L (n = 0, f 1, f 2 , . . .), then the L2[ -n, n] excess E of the system {eiAn'} satisfies -(4L ++)< E S 4L

+ f.

Chapter 3

21 7

(The first result of this type is due to Paley and Wiener [1934, p. 941.) Problem 4. Redheffer [ 1977, p. 451.

Section 3 Theorem 5. This result is implicit in the proof of Theorem XIX in Levinson [ 19401.

Section 4 Theorems 6, 7. Levinson [ 1936, 19401. Theorem 8. See, e.g., Alexander and Redheffer [1967, Remark 3, p. 611. Theorems 9, 10. Schwartz [1959, p. 1021. Problem 1. This result was proved by Paley and Wiener [1934, p. 891 under the added assumption that lim l n / n = 1 . That this condition must in fact hold whenever the system {eiAn'}is exact is a consequence of a wellknown property of entire functions (see, e.g., Levinson [1940, p. 251).

Section 5 Theorem 11. Redheffer [1957]. The theorem remains valid for complex sequences (Alexander and Redheffer [1967]; cf. Problem 1). When p = 2 and when the A,,'s are regularly distributed, then more can be said. Suppose that { A n } is a real sequence such that the number of points of the sequence in the interval ( t ,t I) is uniformly bounded for t E (-a, 'm). Then each of the following conditions implies that the systems {eiAn'}and (eipPi') have the same excess in L 2 [ - A, A] :

+

1.

ClA, I%, - p,I

< w for some positive number

/in[,'

s;

cn/n < co 2. S c,, where t;, decreases to zero and (Sedleckii [1977b]; it is also shown that the condition A, - p n + 0 is not sufficient to guarantee the equality of the excesses.) For further results in this direction, see Sedleckii [1974, 1977bI and Redheffer [1977]. Although Theorem 11 is much weaker than Theorem 4 when pn = n

218

Notes and Comments

( n = 0, f 1, k2,. . .), the convergence criterion is sharp in the following sense: If { E , } is an arbitrary sequence of positive numbers for which infe, = 0, then there exist real separated sequences {A,} and { p , } such that

(Peterson [1974]). Here E(1) and E ( p ) denote the respective excesses of JI e i A n and f } (e""} in L 2 [ -n, n].

Theorem 12. Elsner [1971]. The theorem was rediscovered by Young [1976a], and it is this proof that we have reproduced here. Recently Sedleckii [1978] has shown that in the spaces L ' [ - A , A] and C[ - A , A] the conclusion of the theorem is no longer true: Put 1, = n ( n = 0, k 1, f 2 , . . .) and p, = n for n 5 0, p, = n i( - 1)"h for n > 0 ( h is real and nonzero). Then the system { e i p n fhas } excess 1 in L ' [ -n, 711, while in C[ -n, n] it is exact. Notice of course that the trigonometric system is exact in L ' [ -n, 713 and has deficiency 1 in C[ -n, n]. Whether or not the theorem remains valid in Lp, 1 < p < co, is an open problem.

+

Problem 1. Redheffer [ 1977, p. 111. The references for Theorem 1 1 apply here as well.

Section 6 Theorem 14. Levinson [1935]. For a more elementary proof, see Redheffer [1977, p. 221. Theorem 15. Schwartz [1959]. The theorem remains valid for complex 2,'s (Redheffer [1968, Remark 2, p. 1051). Problem 9. Redheffer [1954, p. 601

For an excellent account of the work of Beurling and Malliavin, see Kahane [1966] and Redheffer [1977]. (See also Koosis [1979].) When a sequence (An}?ao of real or complex numbers is sufficiently regular, in the sense that sup,(1, - nl < co, the corresponding completeness radius is equal to n. The situation is dramatically different when completeness over a union of intervals is considered. Theorem (Landau [19641):

219

Chapter 4

Given E > 0, there exists a real symmetric sequence {A,l}?,x, with - nl < E such thut the system { e i L n l is } complete in C(S), where S is any finite union of intervals qf the form It - 2nnI < IT - 6, with n an integer and 6 > 0.

CHAPTER 4 Section 1 Moment problems. The term moment was introduced into mathematical terminology by Stieltjes [1894]. In this classic paper the following problem, which he called the problem of moments, is stated and completely solved: Find a bounded nondecreasing function p(t) on the interval [0, a) such that its "moments" jp t" d p ( t ) , n = 0, 1,2,. . ., have prescribed values

:{

n

t"dp(t) = c , ,

= 0,

I, 2,. . ..

Subsequent variants of the problem include the Hamburger moment problem (Hamburger [1920]), in which the interval [0, co) is replaced by ( - m, XI); the Huusdorff' moment problem ([Hausdorff [1923]), in which [0, ac) is replaced by [0, I]; and the trigonometric moment problem

jo2n

ei"'dp(t) = c,,

n

=

0, f 1, f 2 , . . .,

considered by Akhiezer and Krein [1934]. For a detailed study of these and other moment problems, see Shohat and Tamarkin [1943] and Akhiezer [19651. A different approach, based on the spectral theorem for selfadjoint operators, is found in Dunford and Schwartz [1963, pp. 125&1256]. Other references: Akhiezer and Krein [1962], Natanson [1965, pp. 1261701, and Krein and Nudelman [1977]. Theorem 1. Paley and Wiener [1934, p. 891, under the additional assumption that lim ),,,in = 1. The proof in the text is from Young [1979]. Example 3. This example appears, in essence, in Paley and Wiener [ 1934, pp. 114-~1161. For additional examples and applications of Lagrange type series, especially to uniqueness and growth theorems, see Boas [ 1954, Chaps. 9, 101 and Levin [1964, Chap. IV]; see also Section 5.

220

Notes and Comments

Theorem 2. Hahn [1927, p. 2161 proves a slightly more general result when H is a Banach space. For this and related results, see Dunford and Schwartz [1958, pp. 86-87].

Section 2 Bessel sequences and Riesz-Fischer sequences were introduced and extensively studied by Bari [19511. Her terminology is slightly different from OUTS.

Theorem 3. The original form of this theorem, with H = L2[0, 11, is due to Boas [1941]. It was rediscovered by Bari [1951], where a number of other equivalent conditions are also given. Interpolation problems in spaces of analytic functions. See Carleson [19581, Shapiro and Shields [ 19611, Hoffman [1962, pp. 194-2081, Bochner [19641, Rosenbaum [1967, 19681, Levin [1969], Duren [1970, Chap. 91, Earl [ 19701, Sedleckii [ 1971b, 19731, Binmore [ 19721, Duren and Williams 19721, Taylor and Williams [1972], Levin and Lyubarskii [1974, 19751, vedenko [1974], Young [1974a, 1976b1, Svedenko and Turku [19751, and Neville [19771.

f

Problem 5. The suggested proof is due to Fejer and Riesz [1921]. The inequality named after them is valid whether the coefficients c, are real or complex, and has important function-theoretic applications (see, e.g., Duren [ 19701). For additional proofs of Hilbert's inequality, together with applications, generalizations, and historical remarks, see Hardy, Littlewood, and Polya [1952, Chap. 1x1. Problem 8. See, e.g., Young [1974a].

Section 3 Theorem 4. Titchmarsh [ 19281. Many proofs have been given; see, e.g., Paley and Wiener [ 1934, p. 1171, Ingham [ 19361, and Amerio [ 19411. The following more general result, which extends the Hausdorff-Young theorem, appeared without proof in Titchmarsh [ 19251 and was subsequently proved in Titchmarsh [1965]: If {A,} is a separated sequence of real numbers, if p >= 2, and if {c,,} E then the series cneiAn'converges in the mean of order p over every finite interval. The 1"'s may be complex

1

221

Chapter 4

provided they are separated and sup,IImA,l < co (see, e.g., Young [1974b]). For a discussion of various other types of convergence of nonharmonic Fourier series, see Kac [19411, Bellman [1943], and Cossar [1975/76]. Open problem: Let {A,} be a sequence of real numbers such that Clf(A,)12 < 00 for every f E P. Is it possible to partition {A,} into a finite number of separated sequences?

Theorem 5. Both the theorem and the counterexample following it are due to Ingham [1936]. A similar but less precise result had already been obtained by Paley and Wiener [1934, p. 1181. Problem 3. Ingham [ 19361. Problem 4. Young [1976c, p. 3171.

Section 4 Theorem 7. Gurevich [ 1941, Theorem 31, under additional hypotheses. The proof in the text is from Singer [1970, Part I, Theorem 8.ldI. Theorem 8. Put another way, this is simply the assertion that a system of vectors in a separable Hilbert space is a Riesz basis if and only if it possesses a biorthogonal system which is also a Riesz basis. The formulation in the text appears explicitly in Korobeinik [1976]. Problem 3. Grinblium [1948]. See, e.g., Singer [1970, p. 751.

Section 5 Interpolation in the Paley- Wiener space was studied extensively by Young [1974a]. For additional facts about functions of sine type, see Levin [1961, 19691, and Sedleckii [1970, 1972al. Theorem 10. Levin [1961] was the first to prove that the system is a basis for L * [ - x , n]. Golovin [1964] showed that it is in fact a Riesz basis. The proof given in the text is due essentially to Levin [1969]. For an impressive generalization of Levin’s theorem, see Katsnelson [19711 (see also Avdonin

222

Notes and Comments

[19741). A generalization in another direction has been given by Levin and Lyubarskii [19741. The duality between basis theory and interpolation theory has been exploited in many different directions. The following papers deserve mention: Binmore [19721, Dragilev, Zaharjuta, and Korobeinik [19741, Levin and Lyubarskii [1975], Korobeinik [1976, 19793, Oskolkov [19783, and Martirosjan [19791. Problem 3. Sedleckii [19701. Problem 7. This result is due to Levin [19691, who shows that the mapping lP+ E : thus defined is onto E:.

Section 6 Proposition 3 and Corollary 1. Young [1974al. Lemma 3. Duffin and Schaeffer [1952]. Lemma 4. This is a special case of a theorem of Bade and Curtis [1966, p. 3931. The elementary proof given in the text is from Stray [1972]. Theorem 11. Young [1976b]. Open problem: Let {A,,} and { p n }be two sequences of points lying in a fixed horizontal strip and suppose that Re I,,= Rep,, for all n. If {I,}is an interpolating sequence for the Paley-Wiener space P, does it follow that {p,} is also an interpolating sequence for P?

Section 7 The substance of this section is taken from Duffin and Schaeffer [19521.

Section 8 Theorem 13. Duffin and Schaeffer [1952]. Corollary. Young [1975bl.

223

Chapter 4

Open problem: Does the corollary to Theorem 13 remain valid if the term “Riesz basis” is replaced by “Schauder basis” in both the hypothesis and the conclusion? Theorem 14. Duffin and Schaeffer [1952]. Corollary 1. Young [1974a, p. 1081. Corollary 2. This result is stated, with an incorrect proof, in Kadec [19641, in which a theorem of Duffin and Schaeffer is misquoted. Katsnelson [I9711 has proved an even stronger result, which contains both Corollary 2 and Theorem 10 as special cases (see also Avdonin [1974]). The proof in the text is due to Young [1975a]. Recently, Pavlov [1979] has obtained a striking criterion-based on the “Muckenhoupt condition”-for a system of complex exponentials to form a Riesz basis for L 2 [ --A, -A].Using that criterion, HruSCev [I9791 has also derived both Corollary 2 and Theorem 10-and more. Open problem: Is the system {e*i(n-*)f

a basis for L2[-n,n]?

Problem 2. Young [1975b]. A stronger result appears in Young [1976c].

Section 9 Theorem 15. Duffin and Schaeffer [1952]. The first result of this sort is due to Paley and Wiener [1934, p. 1131, under the assumption that each 1, is real and sup,)l, - n( < 1/n2. (Cf. Walsh [1921], where it is shown that if A,, is “very near” to n, then each function in L2[0,n] has both an a, cos nt and a nonharmonic Fourier ordinary Fourier cosine series cosine series b, cos Ant, which are uniformly equiconvergent on [0, -A].) The following impressive generalization for Lp is due to Sedleckii [1971al.

E=

Theorem. If {A,} is a sequence of real or complex numbers such that

for some p , 1 < p

5 2, then the set { e i L n fis} complete in L~[--A,-A]and

224

Notes and Comments

possesses a unique biorthogond set { h,(t)} in L4[ -n, n] such that for every f E LP[ - R , n] the series

converges to zero uniformly on every compact subset of ( - n,n). The theorem above generalizes a classical result of Levinson [1940, Chap. IV], in which the An's are assumed to satisfy the more restrictive condition [An - nl 5 L < (p - 1)/2p (n = 0, f1, f 2 , . . .). Levinson showed that his result is sharp in the sense that if L = (p - 1)/2p, then the conclusions of the theorem no longer hold. For a variety of other equiconvergence results, in which the A,,% are the zeros of an entire function belonging to a special class (for example, the class of functions of sine type), see Sedleckii [1970, 1972a, 1972b, 1975a, 1975b, 1977al (see also Verblunsky [1956]). Problem 4. Duffin and Schaeffer [19521.

Ahlfors, L. V. [1979]. “Complex Analysis,” 3rd ed. McGraw-Hill, New York. Akhiezer, N. 1. [1956]. ”Theory of Approximation.” Ungar, New York. Akhiezer, N. I. (19651. “The Classical Moment Problem.” Hafner, New York. Akhiezer, N. I.. and Krein, M. G. [1934]. Ueber Fouriersche Reihen beschrankter summierbarer Funktionen und ein neues Extremumproblem. I, 11. Commun. SOC.Math. Kharkoff 9, 9-28; 10,3-32. Akhiezer. N. I., and Krein, M. G. [1962]. “Some Questions in the Theory of Moments,” Translations of Mathematical Monographs, Vol. 2. Am. Math. SOC.,Providence, Rhode Island. Alexander, W. O., Jr., and Redheffer, R. M. [1967]. The excess of sets of complex exponentials. Duke Math. J. 34, 59-72. m x . Sc. Norm. Super. Amerio, L. [1941]. Sulla convergenza in media della serie ~ , a n e ‘ A mAnn. 0 Pisa 10, 191-198. Anderson, J. M. [ 19781. Muntz-Szasz theorems and lacunary entire functions. Linear Spaces Approx., Proc. Conf., Oberwolfach, 1977. Int. Ser. Numer. Math. 40, 491-501. Aronszajn, N. [1950]. Theory of reproducing kernels. Trans. Am. Math. SOC.68, 337-404. Arsove, M. G. [1958]. Similar bases and isomorphisms in Frkchet spaces. Math. Ann. 135, 283-293. Arutjunjan, F. G. [1972]. Bases of the spaces L,[O, I] and c[O, I]. Math. Notes 11, 152-157. Avdonin, C. A. [1974]. On the question of Riesz bases of exponential functions in L2. Vesfn. Leninyrad Wniv. Ser. Mat., No. 13, 5-12. Babenko, K. I. [1948]. On conjugate functions. Dokl. Akud. Nauk SSSR 62, 157-160. Bade, W. G., and Curtis, P. C., Jr. [1966]. Embedding theorems for commutative Banach algebras. Par. J. Math. 18, 391-409. Banach, S. [I 9321. “Theorie des opkations lineaires.” Warsaw. Reprinted by Chelsea, Bronx, New York, 1955. Bari, N. K. [1951]. Biorthogonal systems and bases in Hilbert space. Ufen Zap. Mosk. Gos. Unio. 148, Mat. 4, 69-107. Bellman, R. [1943]. Convergence of non-harmonic Fourier series. Duke Math. J . 10, 551-552. Berberian, S. K. [1961]. “Introduction to Hilbert Space.” Oxford Univ. Press, London and New York. Bergman, S. [1947]. “Sur les fonctions orthogonales de plusieurs variables complexes avec les applications h la theorie des fonctions analytiques.” Gauthier-Villars, Paris. Bergman, S. [1950]. ”The Kernel Function and Conformal Mapping.” Am. Math. SOC.,New York. 225

226

References

Bernstein, S. N. [1912a]. Demonstration du theoreme de Weierstrass fondee sur le calcul de probabilite. SoohSi.. Har’kou. M a t . ObSE. ( 2 ) 13, 1-2. Bernstein, S. N. [1912b]. Sur I’ordre de la meilleure approximation des fonctions continues par des p o l y n h e s de degre donne. Doctoral Dissertation (French translation). Menr. de I’Academie R(J.vuI~ de Belyique. C1.Sci.. Collection (2) 4, 1 103. Bernstein, S. N. [1923]. Sur une propriete des fonctions entieres. C. R. Accrd. Sci. 176, 16031605. Besicovitch. A. S . [1932]. “Almost Periodic Functions.” Cambridge Univ. Press, London. Bessaga, C.. and Pelczynski, A. [1958]. On bases and unconditional convergence of series in Banach spaces. Stud. Math. 17. 151-164. Beurling, A,, and Malliavin, P. [1967]. On the closure of characters and the zeros of entire functions. Acta Mtrtk. 118, 79-93. Billard, P. [1972]. Bases dans H et bases de sous-espaces de dimension finie dans A. Lincwr Operators Appro.u.. Proc. Con/:. Oherblo(/uCh, 1971. Int. Ser. Numcr. Math. 20, 310- 324. Binmore, K . G . [1972]. Interpolation, approximation, and gap series. Proc. London Muth. SOC.25, 751-768. Birkhoff, G . [1935]. Orthogonality in linear metric spaces. Duke Math. J . 1, 169-172. Birkhofl, G., and Rota, G. C. (19601. On the completeness of Sturm-Liouville expansions. Am. Math. Mon. 67, 835-841. Birkhoff, G . . and Rota, G . C. [1962]. “Ordinary Differential Equations.” Ginn. New York. Boas, R. P., Jr. [1937]. The derivative of a trigonometric integral. J . London Math. SOC. 12, 164- 165. Boas, R. P., Jr. [1938]. Representations for entire functions of exponential type. Ann. Muth. 39, 269-286 (correction, 40, 948). Boas, R. P., Jr. [1940]. Expansions of analytic functions. Trans. .4m.Math. SOC.48, 467- 487. Boas, R. P., Jr. [1941]. A general moment problem. Am. J . Math. 63, 361-370. Boas, R. P.. Jr. [1952]. Integrability along a line for a class of entire functions. Trans. Am. Math. Soc. 73, 191-197. Boas, R. P., Jr. [1954]. “Entire Functions.” Academic Press, New York. Boas, R. P.. Jr. [1955]. Isomorphism between H p and L p . Amer. J . Math. 77, 655-656. Boas, R. P., Jr. [1972]. Summation formulas and band-limited signals. Toliokir Muth. J . 24, I21 125. Boas, R. P., Jr., and Pollard, H. [1948]. The multiplicative completion of sets of functions. Bull. Am. Math. Soc. 54, 518-522. Bochner, S. [1964]. Interpolation of general bounded and of almost periodic sequences by functions of stratified exponential type. Proc. Natl. Acad. Sci. U . S . A . 51, 164 168. Botkarev, S. V. [1974]. A basis in the space of functions that are continuous in the closed disk and analytic in its interior. SOO.Math. Dokl. 15, 1195- 1198. Bohr. H. [1925a]. Zur Theorie der fastperiodischen Funktionen. Acta Math. 45, 29- 127. Bohr, H. [3925b]. Zur Theorie der fastperiodischen Funktionen. Actu Muth. 46, 101-214. Bohr, H. [1926]. Zur Theorie der fastperiodischen Funktionen. Acta Math. 47, 237-281. Bohr. H . [1947]. “Almost Periodic Functions.” Chelsea, Bronx, New York. Borel, E. [1897]. Sur les zeros des fonctions entieres. Acta Math. 20, 357 396. Borel. E. [1921]. “Lecons sur les fonctions entieres.” Deuxieme edition revue et augmentee d’une note de M.G . Valiron. Gauthier-Villars, Paris. Bourgin. D. G . [1946]. A class of sequences of functions. Tram. Am. Murk SOC.60, 478 518. BratiSEev, A. V., and Korobeinik, Ju. F. [1976]. The multiple interpolation problem in the space of entire functions of given proximate order. Math. USSR-lzo. 10, 1049-1074. Brauer, F. [1964]. The completeness of biorthogonal systems. Mirh. M a h J . 11. 379 383. -

-

227

References

Byrnes, J. S. [ 1972). Functions which multiply bases. Bull. London Murh. Sot,. 4, 330--332. Carleman. T. [ 19221. Uber die Approximation analytischer Funktionen durch lineare Aggregate von vorgegebene Potenzen. Ark. M u / . Asrron. Fvs. 17(9), 1-30, Carleson, L. [ 195x1. An interpolation problem for bounded analytic functions. Am. J . Mtrrli. 80, 921 930. Carleson, L. (19661. On convergence and growth o f partial sums of Fourier series. Acrtr Mtrrli. 116, 135-157. Cartwright. M. L. [1935]. On certain integral functions of order I and mean type. Proc.. Cumhriclp P h i k ~ . Suc. ~ . 31, 347-350. Cartwright. M. L. [ 19561. ”Integral Functions.” Cambridge Univ. Press, London. Ciesielski. Z. 119631. Properties of the orthonormal Franklin system. Srud. M u r k . 23, 141- 157. Ciesielski. Z. [1966]. Properties of the orthonormal Franklin system. 11. Srud. Murh. 27, 2x9323. Ciesielski. Z.. and Domsta. J. [1972]. Construction of an orthonormal basis in C””(1“) and U;?f‘)). Stud. hfu/lr. 41, 211-224. Ciesielski. Z.. Simon, P.. and Sjiilin, P. [1977]. Equivalence of Haar and Franklin bases in L,, spaces. S/rrcl. Morh. 60. 195 210. Clarkson, J. A,, and Erdiis, P. [1943]. Approximation by polynomials. Duke Muth. J . 10, 5-11,

Cook, T. A. [1970]. On normalized Schauder bases. Ann. Muth. Mon. 77, 167. Cossar, J. [ 1975/76]. Mean-square convergence of non-harmonic trigonometrical series. Proc,. R . SO(,.Ediiihurqlt. 75A. 24. 297 323. Dalzell, D. P. [ 19451. On the completeness of a series of normal orthogonal functions. J . London MU/h. S O C . 20, 87--93. Davie. A. M. [1973]. The approximation problem for Banach spaces. Bull. London M u r h . Sot,. 5, 261 266. Davis, P. J. [1963]. “Interpolation and Approximation.” Ginn (Blaisdell), New York. Davis. W. J. [1969]. Basis preserving maps. Proc. Anr. Mtirlr. Soc. 22, 34-36. de Branges. L. [ 19681. ”Hilbert Spaces of Entire Functions.” Prentice-Hall, Englewood Cliffs, New Jersey. Deutsch. F.. and Morris. P. [1969]. On simultaneous approximation and interpolation which preserves the norm. J . appro.^. Tlreurj, 2, 3.55-373. Diestel. J . [1975]. ”Geometry of Banach Spaces -Selected Topics,” Lecture Notes in Mathematics, No. 485. Springer-Verlag, Berlin and New York. Dragilev. M. M.. Zaharjuta, V. P., and Korobeinik. J. F. (19741. A dual connection between certain questions of basis theory and interpolation theory. Sou. Muth. Dokl. 15, 533-537. Duffin, R. J . . and Eachus. J. J. [1942]. Some notes on an expansion theorem of Paley and Wiener. Bull. Am. Mtrrh. Soc. 48, 850-855. Duffin. R. J.. and Schaeffer. A. C. [1938]. Some properties of functions of exponential type. Bull. .4m. Mrr/li. SUC.44. 236-240. Duffin. R. J.. and SchaeKer. A. C. [1945]. Power series with bounded coefficients. A m . J . Mtr/lt. 67, 141-154. Duffin. R. J . , and SchaeKer. A. C. [1952]. A class of nonharmonic Fourier series. Truns. A m . Mu/lr. Snr. 72. 341-306. Dunford, N.. and Schwartz, J . T. [1958]. “Linear Operators. Part I : General Theory.” Wiley (Interscience), New York. Dunford, N.. and Schwartz, J. 1’.[1963]. ”Linear Operators. Part 11: Spectral Theory.” Wiley (Interscience). New York. Duren, P. L. [1970]. “Theory of H P Spaces.” Academic Press. New York. -

228

References

Duren, P. L., and Williams, D. L. [1972]. Interpolation problems in function spaces. J. Funct. Anal. 9, 75-86. Dvoretzky, A,. and Rogers. C. A. [1950]. Absolute and unconditional convergence in normed linear spaces. Proc. Natl. Acad. Sci. U . S . A . 36, 192-197. Earl, J. P. [1970]. On the interpolation of bounded sequences by bounded functions. J. London Math. Soc. 2, 544-548. Elsner, J. (19711. Zullssige Abanderungen von Exponentialsystemen im L p ( - A , A ) . Math. z. 120,211-220. Enflo, P. [1973]. A counterexample to the approximation property in Banach spaces. Actci Math. 130, 309-317. Fejer. L.. and Riesz, F. [1921]. Uber einige functionentheoretische Ungleichungen. Math. Z . I I , 305-314. Feller, W. (19661. “An Introduction to Probability Theory and its Applications,” Vol. 11. Wiley, New York. Feller, W. 119681. On Miintz’ theorem and completely monotone functions. A m . Math. Mon. 75, 342-350. Gelbaum, B. R. [1950]. Expansions in Banach spaces. Duke Math. J . 17, 187-196. Gelbaum, B. R. [1951]. A nonabsolute basis for Hilbert space. Proc. A m . Math. Soc. 2, 720721. Gelfand, 1. M. [1951]. Remark on the work of N. K. Bari “Biorthogonal systems and bases in Hilbert space.” W e n . Zap. Mosk. Gos. Univ. 148, Mat. 4, 224-225. Giroux, A,, Rahman, Q. I., and Schmeisser, G. [1979]. On Bernstein’s inequality. Can. J. Malh. 31, 347-353. Gohberg, 1. C., and Krein. M. G. (19691. “Introduction to the Theory of Linear Nonselfadjoint Operators,” Translations of Mathematical Monographs, Vol. 18. Am. Math. SOC., Providence, Rhode Island. Golovin. V. D. [1964]. Biorthogonal expansions in linear combinations of exponential functions in L z . Zap. Har’kov. Gus. Univ. i Har’kov. Mat. Obi? (4) 30, 18-29. Gosselin, R. P., and Neuwirth, J. H. [1968/69]. On Paley-Wiener bases. J. Math. Mech. 18, 871-879. Graves, L. M. [1941]. Some general approximation theorems. Ann. Math. 42, 281-292. Grinblium, M. M. [1948]. On a property of a basis. Dokl. Akad. Nauk SSSR 59, 9-1 I . Grothendieck, A. [ 19551. Produits tensoriels topologiques et espaces nucleaires. M e m . Am. Math. Soc. 16, 1-191. Gurarii, V. I., and Gurarii, N. I. [1971]. Bases in uniformly convex and uniformly flattened Banach spaces. Math. USSR-lzv. 5, 220-225. Gurarii, V. 1.. and Meletidi, M. A. [1970]. Stability of completeness of sequences in Banach spaces. Bull. Acad. Pol. Sci., Ser. Sci. Math. Astron. Phys. 18, 533-536. Gurevich, L. A. [1941]. Sur une propriete de la base dans I’espace de Hilbert. Dokl. Akad. Nauk SSSR 30, 289-291. Hadamard, J. [1893]. Etude sur les proprietks des fonctions entieres et en particulier d u n e fonction consideree par Riemann. J . Math. pures et appliquies 9, 171-215. Hahn, H. [1927]. u b e r lineare Gleichungssysteme in linearen Raumen. J. Reine Angew. Math. 157, 214-229. Halmos, P. R. [1951]. “Introduction to Hilbert Space and the Theory of Spectral Multiplicity.” Chelsea, Bronx, New York. Halmos, P. R. [1967]. “A Hilbert Space Problem Book.” Van Nostrand, New York.

References

229

Hamburger, H. [ 19201. Uber eine Erweiterung des Stieltjesschen Momentenproblems. Math. Ann. 81, 235-319. Hammersley, J. M . [1953]. A nonharmonic Fourier series. Acra Math. 89, 243-260. Hardy, G. H. [ 19491. “Divergent Series.” Oxford Univ. Press (Clarendon), London. Hardy, G . H.. Littlewood. J . E., and Polya. G. [1952]. “Inequalities.” 2nd ed. Cambridge Univ. Press, London. Hausdorff. F. [1923]. Momentprobleme fur ein endliches Intervall. Math. Z . 16, 220-248. Hille, E. [1972]. Introduction to general theory of reproducing kernels. Rocky Mount. J . Math. 2, 321 - 368. Hoffman. K. 119621. “Banach Spaces of Analytic Functions.” Prentice-Hall, Englewood Cliffs. New Jersey. Holland, A. S. B. [1973]. “Introduction to the Theory of Entire Functions.” Academic Press, New York. HruStev, S. V. [ 19791. Perturbation theorems for bases of exponentials and Muckenhoupt’s condition. SOP.Math. Dokl. 20, 665-669. Ingham, A. E. [1936]. Some trigonometrical inequalities with applications to the theory of series. Math. Z . 41, 367-379. James. R. C. [1945]. Orthogonality in normed linear spaces. Duke Math. J . 12, 291-302. James, R. C. [1950]. Bases and reflexivity of Banach spaces. Ann. Math. 52, 518-527. Jensen, J. L. W. V. [1899]. Sur un nouvel et important theoreme de la theorie des fonctions. Acta Marh. 22, 359-364. Johnson, W. B.. Rosenthal. H. P.. and Zippin. M. [1971]. On bases, finite-dimensional decompositions and weaker structures in Banach spaces. Isr. J . Math. 9, 488-506. Kac, M. [ 19411. Convergence and divergence of non-harmonic gap series. Duke Math. J . 8, 541 -545. Kadec. M . I.[1964]. The exact value of the Paley-Wiener constant. Sou. Math. Dokl. 5, 559561. Kahane. J . P. [1958]. Sur la totalite des suites d’exponentielles imaginaires. Ann. Inst. Fourier (Grenohle) 8, 273 -275. Kahane, J. P. [1966]. “Travaux de Beurling et Malliavin.” Benjamin, New York. Karlin, S. (19481. Bases in Banach spaces. Duke. Math. J . 15,971-985. Katsnelson, V. E. [1971]. Exponential bases in L’. Funct. Anal. Appl. 5, 31-38. Knopp, K. [1951]. “Theory and Application of Infinite Series,” 2nd ed. Blackie, Glas, w and London. Koosis, P. [1960]. Sur la totalite des systemes d’exponentielles imaginaires. C. R. Acud. Sci. 250, 2102-2103. Koosis, P. 119793. Proof of the Beurling-Malliavin theorem by duality and harmonic estimation. Harmonic Analysis in Euclidean Spaces, Proc. Symp. Pure Math. XXXV (Part I). Amer. Math. SOC.,Providence, Rhode Island. Korevaar, J. [1947]. A characterization of the submanifold of C[a,61 spanned by the sequence { x n x). Indag. Math. 9, 360-368. Korobeinik. Ju. F. [1976]. Unconditional bases in a Hilbert space. Math. N o r a 19, 153-157. Korobeinik. Ju. F. [ 19791. The moment problem, interpolation, and basicity. Math. USSR-Izu. 13, 277-306. Kiithe. G. [ 19361. Das Triigheitsgesetz der quadratischen Formen im Hilbertschen Raum. Moth. Z . 41, 137- 152. Krein, M. G.. and Liusternik. L. A. [1948]. Functional analysis. In “Thirty Years of Mathe-

230

References

matics in the USSR (1917-1947),” pp. 608-672. Gos. Izdat. Tekhn.-Teor. Lit., Moscow-Leningrad. Krein, M. G., and Nudelman, A. A. [1977]. “The Markov Moment Problem and Extremal Problems,” Translations of Mathematical Monographs, Vol. 50. Am. Math. SOC.,Providence, Rhode Island. Kreip, M. G., Milman, D. P.,and Rutman, M. A. [1940]. On a property of the basis in Banach space. Zap. Mat. T-ua (Kharkou) 16, 106-108. Krotov, V. G. [1974]. Representation of measurable functions by series in the Fabq-Schaudcr system, and universal series. Sou. Math. Dokl. 15, 351-355. Kwapien, S., and Pelczynski, A. (19701. The main triangle projection in matrix spaces and its applications. Studia Math. 34, 43-68. Landau, H. J. [1964]. A sparse regular sequence of exponentials closed on large sets. Bull. Am. Math. SOC.70, 566-569. Lerer, L. E. [1969]. The stability of bases of locally convex spaces. Sou. Math. Dokl. 10, 24-28. Levin, B. Ja. [1961]. On bases of exponential functions in Lz. Zap. Har’kou. Gos. Univ. i Har’kou. Mat. Obi?. ( 4 ) 27, 39-48. Levin, B. Ja. [1964]. “Distribution of Zeros of Entire Functions,” Translations of Mathematical Monographs, Vol. 5. Am. Math. Soc., Providence, Rhode Island. Levin, B. Ja. [ 19691. Interpolation by entire functions of exponential type. Proc. Phys.-Technol. Inst. Low Temp., Acad. Sci. Ukr. SSR, Math. Phys. Funct. Anal. No. I, pp. 136-146. Levin, B. Ja., and Lyubarskii, J. 1. [1974]. Interpolation by entire functions of exponential type and its application to expansions in series of exponentials. Funct. Anal. Appl. 8, 172- 174. Levin, B. Ja., and Lyubarskii, J. I. [1975]. Interpolation by means of special classes of entire functions and related expansions in series of exponentials. Math. USSR-lzu. 9, 621 -662. Levin, B. Ja., and Ostrovskii, 1. V. [1979]. Small perturbations of the set of roots of sine-type functions. Izu. Akad. Nauk SSSR, Ser. Mat. 43, 87- 110. Levinson, N. [1935]. On the closure of { e ’ l m Xand } integral functions. Proc. Cumhrrdgc Phrlos. 31, 335-346. SCJC. Levinson, N. [1936]. On the closure of {e‘A”x}. Duke Muth. J . 2, 511-516. Levinson, N. [1940]. “Gap and Density Theorems,” Am. Math. Soc. Colloq. Publ., Vol. 26. Am. Math. SOC.,New York. Lin, B.. and Singer, 1. [1971]. On conditional bases of 1’. Comrnenfuf. Muth. Pr. Mat. 15, 135- 139. Lindenstrauss. J., and Pelczynski, A. [1968]. Absolutely summing operators in &-spaces and their applications. Sfud. Math. 29, 275-326. Lindenstrauss, J., and Pelczynski, A. [1971].Contributions to the theory of the classical Banach spaces. J. Funct. Anal. 8, 225-249. Lindenstrauss, J., and Tzafriri. L. [1977]. “Classical Banach Spaces 1.” Springer-Verlag. Berlin and New York. Lindenstrauss, J., and Zippin, M. [1969]. Banach spaces with a unique unconditional basis. J. Funct. Anal. 3, 115-125. Lorch, E. R. [1939]. Bicontinuous linear transformations in certain vector spaces. Bull. A m . Math. Soc. 45, 564-569. Lorentz, G. G. [1953]. ”Bernstein Polynomials.” Univ. of Toronto Press, Toronto. Luxemburg, W. A. J. (19761. Miinti-Szisz type approximation results and the Paley-Wiener theorem. I n ”Approximation Theory, 11” (G. G. Lorentz, C. K. Chui, and L. L. Schumaker, eds.), pp. 437-448. Academic Press, New York.

References

231

Luxemburg, W. A. J . , and Korevaar, J. [1971]. Entire functions and Muntz Sziisz type approximation. Truns. Am. Mu/li. SOC.157, 23 37. Marcinkiewicz, J. [ 19371. Quelques theoremes sur les series orthogonales. Ann. Soc. Pol. Mufh. 16, 84 96. Markushevich, A. I. [ 196.51. “Theory of Functions of a Complex Variable.” Vol. 2. PrenticeHall, Englewood Cliffs, New Jersey. Markushevich. A. 1. [ 19661. “Entire functions.” Elsevier. Amsterdam. Marti. J . 7’. [ 19691. “Introduction to the Theory of Bases.” Springer-Verlag. Berlin and New York. Martirosjan. V. M. [ 19791. Closure and basicity of certain biorthogonal systems and the solution of the multiple interpolation problem in H , [ a ; o ] . Sou. Math. Dokl. 20, 260-263. Meletidi. M. A. [ 19721. Complete systems and bases in the spaces C‘ and L,. Soc. M u f h . Dokl. 13, 1372- 1376. Muntz, C. H. [1914]. ”Uber den Approximationssatz von Weierstrass.” pp. 303- 312. H. A. Schwarr Festschr.. Berlin. Natanson, 1. P. [1965]. ”Constructive Function Theory,” Vol. 2. Ungar. New York. Nehari, Z. [ 1952). “Conformal Mapping.” McGraw-Hill, New York. Neville, C. W. [ 19771. A short proof of an inequality of Carleson’s. Proc. h i . Moth. SOC. 65, 131 132. O’Hard, P. J . [I973]. Another proof of Bernstein’s theorem. A m . M a / h . Mon. 80, 673-674. Orlicz, W. 119331. Uber unbedingte Konvergenz in Funktionenraumen 1. Siutl. Math. 4, 33 37. Oskolkov, V. [1978]. Some bases in spaces of regular functions and their application to interpolation. Mofli. USSR-Sh. 34. 215-134. Paley. R. E. A. C.. and Wiener. N. [1934]. “Fourier Transforms in the Complex Domain,” Am. Math. Soc. Colloq. Publ., Vol. 19. Am. Math. SOC.. New York. Pavlov, B. S. [1979]. Basicity of an exponential system and Muckenhoupt’s condition. Sou. Math. Dokl. 20, 655 -659. Pelczynski, A. [1961]. On the impossibility of embedding of the space L in certain Banach spaces. Colloy. Math. 8, 199-203. Pelczynski, A. [1977]. “Banach Spaces of Analytic Functions and Absolutely Summing Operators,” Regional Conference Series in Mathematics, No. 30. Am. Math. Soc., Providence. Rhode Island. Pelczynski. A., and Singer, 1. [1964]. On non-equivalent bases and conditional bases in Banach spaces. Stud. Moth. 25, 5-25. Peterson, D. R. [1974]. The excess of sets of complex exponentials. Proc. A m . Mufh. SOC. 44, 321-325. Phragmen. E.. and LindelBf, E. [1908]. Sur une extension d’un principe classique de I’analyse et sur quelques propriktts des fonctions monogenes dans le voisinage d’un point singulier. Ac,/rr M a f h . 31. 381 -406. Plancherel, M.. and Polya, G. [I938]. Fonctions entieres et integrales de Fourier multiples . 110- 163. (Seconde partie). C’onimenfuf.MufIi. f f e h %10, Pollard, H . , and Shisha. 0. [1972]. Variations on the binomial series. A m . Mufh. Mon. 79, 495-499. Polya. G . [ 19291. Unkrsuchungen uber Lucken und Singularitiiten von Potenzreihen. Murh. Z . 29, 549 640. Polya, G . [1931]. Problem 108. Juhresher. D/sch. Math. Ver. 40, Part 2, 81. Polya, G . . and Szegii. G . [1925]. “Aufgaben und Lehrsltze aus der Analysis,” Vols. I and 11. Springer-Verlag, Berlin and New York.

232

References

Povzner, A. [1949]. On the completeness of the sequence of functions {eiAn’}in L2( -n, K). Dokl. Akad. Nauk SSSR 64, 163-166. Redheffer, R. M. [1953]. On a theorem of Plancherel and Polya. Pac. J . Math. 3, 823-835. Redheffer, R. M. [1954]. On even entire functions with zeros having a density. Trans. Ani. Math. Sue. 11, 32-61. Redheffer, R. M. [1957]. Ganze Funktionen und Vollstindigkeit. Oesrerr. Akad. Wiss. 6, 96-99. Redheffer, R. M. [1968]. Elementary remarks on completeness. Duke Math. J . 35, 103- 116. Redheffer, R. M. [1977]. Completeness of sets of complex exponentials. Adv. Math. 24, 1-62. Retherford, J . R., and Holub, J. R. [1971]. The stability of bases in Banach and Hilbeit spaces. J . Reine Anyew. Math. 246, 136-146. Riesz, F., and Sz.-Nagy, B. [1955]. “Functional Analysis.” Ungar. New York. Rosenbaum, J. T. [1967]. Simultaneous interpolation in H Z . Mich. Math. J . 14, 65-70. Rosenbaum, J. T. [1968]. Simultaneous interpolation in H , , 11. PUC.J . Math. 27, 607-610. Royden, H. L. [1968]. “Real Analysis,” 2nd ed. Macmillan, New York. Ruckle. W. H. (19721. The extent of the sequence space associated with a basis. Can. J. Math. 24, 636-641. Rudin, W. [1966]. “Real and Complex Analysis.” McGraw-Hill, New York. Russell, D. L. [1967]. Nonharmonic Fourier series in the control theory of distributed parameter systems. J. Muth. Anal. Appl. 18, 542-560. Schafke, F. W. [1949]. u b e r einige unendliche lineare Gleichungssysteme. Math. Nuchr. 3, 40-58. Schauder, J. [ 19271. Zur theorie stetiger Abbildungen in Funkionalraumen. Math. 2. 26, 47-65, 417-431. Schonefeld, S. [1972]. Schauder bases in the Banach spaces C k ( P ) .Trans. Am. Muth. SOC. 165, 309-318. Schwartz, L. [1943]. Approximation d’une fonction quelconque par des sommes d’exponentielles imaginaires. Ann. Fac. Sci. Toulouse 6, I 11- 176. Schwartz, L.[ 19591. “Etude des Sommes d’Exponentielles,” Publications de I’lnstitut de Mathematique de I’Universite de Strasbourg, 2nd ed., Vol. V. Hermann, Paris. Sedleckii, A. M. [1970]. On functions periodic in the mean. Marh. USSR-lzv. 4, 1406-1428. Sedleckii, A. M. [1971a]. Nonharmonic Fourier series. Sib. Math. J . 12, 793-802. Sedleckii, A. M. [1971b]. Interpolation by entire functions of exponential type. Tr. Musk. Orden. Lenin. Energ. Inst. No. 89, pp. 55-64. Sedleckii, A. M. [1972a]. On biorthogonal expansions in exponential functions. Muth. U S S R IZV. 6, 579-586. Sedleckii, A. M. [1972b]. Continuation periodic in the mean and bases of exponential functions in Lp (-n, n). Math. Notes 12, 455-458. Sedleckii, A. M. [1973]. Equivalent sequences in certain function spaces. Izo. Vyssh. Uchebn. Zaued., Mat. 134 (No. 7), 85-91. Sedleckii, A. M. [1974]. The stability of the completeness and minimality in L2 of a system of exponential functions. Math. Notes 15, 121-124. Sedleckii, A. M. [1975a]. Equiconvergence and equisummability of nonharmonic Fourier expansions with ordinary trigonometric series. Math. Notes 18, 586- 59 1. Sedleckii, A. M. [1975b]. Expansions in exponential functions. Sib. Math. J. 16, 628-635. Sedleckii, A. M. [1977a]. On a class of biorthogonal expansions in exponential functions. Math. USSR-IZV.11, 375-395. Sedleckii, A. M. [1977b]. Excesses of systems of exponential functions. Math. Notes 22, 941947.

References

233

Sedleckii. A. M. (19781. On completeness of the systems {exp(ix(n + i h ” ) ) } . Anal. Math. 4, 125- 143. Sengupta, A., and Karnick, H. [1978]. Analysis of the neutron slowing down equation. J . Muth. Pliys. 19, 2563-2569. Shah, S. M. [1941]. A note on the classification of integral functions. Math. Student 9,63-67. Shannon. C. E. (19491. Communication in the presence of noise. Proc. I.R.E. 37, 10-21. Shannon. C . E.. and Weaver, W. [1964]. “The Mathematical Theory of Communication.” Univ. of Illinois Press, Urbana. Shapiro, H. S. 119711. “Topics in Approximation Theory,” Lecture Notes in Mathematics, Vol. 187. Springer-Verlag, Berlin and New York. Shapiro, H. S.. and Shields, A. L. [1961]. On some interpolation problems for analytic functions. Am. J . Muth. 83, 513- 532. Shohat. J . A., and Tamarkin, J. D. [1943]. “The Problem of Moments,” Math. Surveys, Vol. I . Am. Math. SOC.. New York. Singer. 1. [1970]. ”Bases in Banach Spaces 1.” Springer-Verlag, Berlin and New York. Sjolin, P. [1977]. The Haar and Franklin systems are not equivalent bases in L’. Bull. Acad. Pol. Sci.. Ser. Scr. Math. Astron. Phys. 25( 1 I), 1099- 1100. Stepanoff. W. [ 19251. Sur quelques generalisations des fonctions presque periodiques. C. R. Acud. Sci. 181, 9 0 ~ ~ 9 4 . Stieltjes, T. J. [1894]. Recherches sur les fractions continues. Ann. Fuc. Sci. Toulouse 8, J I 3122; 9, AS- A47. Stray. A. [1972]. Approximation and interpolation. Pac. J . Muth. 40, 463-475. Svedenko. S. V. [ 19741. Interpolation in certain Hilbert spaces of analytic functions. Math. Notes 15. 56 61. Svedenko, S. V.. and Turku, H. [1975]. Interpolation in the class H2 in a half-plane. Sor. Math. Dokl. 16, 1547--1550. Szasz, 0. [1916]. Ober die Approximation stetiger Funktionen durch lineare Aggregate von Potenzen. Math. Ann. 77, 482-496. Szisz. 0. [1933]. Solution to problem 108. Juhresher. Dtsch. Muth. Ver. 43, Part 2, 20-23. Szisz. 0. [1944]. “Introduction to the Theory of Divergent Series.” University of Cincinnati, Cincinnati. Szego, G . [1939]. “Orthogonal Polynomials.” Am. Math. SOC.,New York. Talaljan, F. A. [1972]. Rearrangements of series in a Hilbert space. Math. Notes 12, 599-602. Taylor. A. E. [1958]. “Introduction to Functional Analysis.‘’ Wiley, New York. Taylor, B. A., and Williams. D. L. (19721. Interpolation of lq sequences by H P functions. Proc. Am. Muth. Soc. 34, 181-186. Titchmarsh, E. C . [1925]. A theorem on trigonometrical series. Proc. London Math. Soc. XXIV xxii-xxiii. Titchmarsh, E. C. [1926]. The zeros of certain integral functions. Proc. London Math. Soc. 25, 283 302. Titchmarsh, E. C. [1928]. A class of trigonometrical series. J . London M a f h . Soc. 3, 300-304. Titchmarsh, E. C . 119391. “The Theory of Functions.” 2nd ed. Oxford Univ. Press, London. Titchmarsh, E. C. [1946]. “Eigenfunction Expansions Associated with Second-order Differential Equations.” Clarendon Press, Oxford. Titchmarsh. E. C . [1965]. On the mean convergence of trigonometrical series. J . London Math. Soc. 40, 594-596 (addendum, 43, 367). Valiron, G . [1949]. “Lectures on the General Theory of Integral Functions.” Chelsea, Bronx, New York. Verblunsky, S. [1956]. On an expansion in exponential series. Q.J . Math. 7 , 231-240.

234

References

Vitali, G. [1921]. Sulla condizione di chiusura di un sistema di funzioni ortogonali. Atti. H. Accud. N E . Lincei, Rend. CI. Sci. Fis., Mut. Not. 30, 498-501. Wallen, L. J. (19691. A stability theorem for frames in Hilbert space. Mich. Muth. J . 16, 149151.

Walsh, J. L. (19211. A generalization of the Fourier cosine series. Truns. Am. Math. Sue. 22, 230- 239. Walsh. J. L. [1935]. "Interpolation and Approximation by Rational Functions in the Complex Domain," Am. Math. SOC.Colloq. Publ., Vol. 20. Am. Math. SOC.,New York. Weierstrass. K. [ I 8761. Zur Theorie der eindeutigen analytischen Funktionen. Ahli. K. Akird Wiss.11-60 (see Werke 2, 77-124, 1895). Weierstrass, K. [I 8851. Uber die analytische Darstellbarkeit sogennanter willkiirlicher Funktionen einer reellen Verlnderlichen. Berl. Ber. 633-640; 789-806 (see Werke 3, 1-37, 1903). Weierstrass, K. [1895]. "Mathematische Werke," 7 vols. [1894-19271. Mayer & Miiller, Berlin. Whittaker. E. T. [1915]. On the functions which are represented by the expansions of the interpolation theory. Proc. R. Soc. Edinburgh 35, 181- 194. Whittaker, J. M. [1935]. "lnterpolatory Function Theory," Cambridge Tracts in Mathematics and Mathematical Physics, No. 33. Cambridge Univ. Press, London. Wolibner. W. [1951]. Sur un polynbme d'interpolation. Culluq. Muih. 2, 136- 137. Young, R. M. [1974a]. Interpolation in a classical Hilbert space of entire functions. Truns. A m . Math. Soc. 192, 97- 114. Young, R. M. [1974b]. An extension of the Hausdorff-Young theorem. Pruc. Am. Math. Sue. 45, 235-236. Young, R. M. [1975a]. A note on a trigonometric moment problem. Proc. Am. Marh. Soc. 49,411-415. Young, R. M. [1975b]. On perturbing bases of complex exponentials in L z ( - R . r). Pror. Am. Math. Soc. 53, 137-140. Young, R. M. [1976a]. A perturbation theorem for complete sets of complex exponentials. Pruc. A m . Mutli. Soc. 55, 318-320. Young, R. M. [1976b]. Interpolation for entire functions of exponential type and a related trigonometric moment problem. Proc. Am. Math. Soc. 56, 239-242. Young, R. M. [1976c]. Some stability theorems for nonharmonic Fourier series. Pruc. A m . Mufh. Soc. 61, 315-319. Young, R. M. [1979]. On a completeness theorem of Paley and Wiener. Proc. Am. Math. SUC.76, 349-350. Zygmund, A. [1959]. "Trigonometric Series," 2nd ed. Cambridge Univ. Press. London.

LIST OF SPECIAL SYMBOLS

A A2

26

B,

84

E:

99

10

P H* Hz(D)

10 174

x

106

Iy

b1 .

x

235

13 21 20

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Numbers in italics refer to pages on which the references are listed.

A

Brauer, F., 210, 226 Byrnes, J. S., 227

Ahlfors, L. V., 60, 225 Akhiezer, N. 1.. 3, 84, 213, 219,225 Jr., 217,225 Alexander. W. 0.. Amerio, L., 220, 225 Anderson, J. M.. 213,225 Aronszajn, N., 206,225 Arsove, M. G., 208,225 Arutjunjan, F. G., 204,225 Avdonin, C. A,, 221, 223, 225

C Carleman, T., 114, 213, 227 Carleson, L., 9, 160, 197, 201, 220, 227 Cartwright, M. L., 210, 212, 227 Ciesielski, Z., 204, 208, 227 Clarkson, J. A,, 213,227 Cook, T. A., 207,227 Cossar, J., 221,227 Curtis, P. C., Jr., 222, 225

B Babenko, K. I., 36,208,225 Bade, W. G., 222,225 Banach, S.. 2, 203, 207, 225 Bari, N. K., 146, 208-210,220,225 Bellman, R., 221, 225 Berberian, S. K., 206, 225 Bergman, S., 206,207,225 Bernstein, S. N., 3, 84, 205, 212. 213, 226 Besicovitch, A. S., 14. 206, 226 Bessaga, C., 203, 208,226 Beurling, A,, 142, 218,226 Billard, P., 204,226 Binmore, K. G., 220, 222,226 Birkhoff, G., 206, 210,226 Boas, R. P., Jr., 101, 204,207,209,210,212215, 219, 220. 226 Bochner, S., 220, 226 BoEkarev, S. V., 204, 226 Bohr, H., 14, 206.226 Borel, E., 54, 210, 21 I , 226 Bourgin, D. G., 209. 226 BratiSCev, A. V., 226

D Dalzell, D. P., 206, 227 Davie, A. M., 205, 227 Davis, P. J., 207, 227 Davis, W. J., 209, 227 de Branges, L., 214, 215, 227 Deutsch. F., 205, 227 Diestel, J., 207, 227 Domsta, J., 204, 227 Dragilev, M. M.,222,227 Duffin, R.J., 44,210,212-214,222-224.227 Dunford, N., 206,219,220, 227 Duren, P.L., 160, 174,204,206,220,227,228 Dvoretzky, A., 204,228

E Eachus, J. J., 44,210,227 Earl, J. P., 220, 228 Elmer, J., 218,228 Enflo, P., 2,205, 228

237

238

Author Index

Erdos, P.. 213, 227 Fejcr, L., 220. 22X Feller, W., 5. 213. 22X C

Gelbaum, B. R., 208,228 Gelfand. 1. M., 208, 228 Giroux. A,. 213, 228 Gohberg, I. C., 46, 228 Golovin, V. D., 221. 228 Gosselin, R. P.,209. 228 Graves, L. M., 228 Grinblium, M. M., 221, 228 Grothendieck, A.. 26, 205, 207, 228 Gurarii. N. 1.. 209, 22N Gurarii, V. I., 209, 228 Gurevich, L. A,. 221, 228 H

Hadamard, J., 54. 66. 74. 21 I. 228 Hahn. H., 220.228 Halmos, P. R.,40, 45, 48. 206, 210, 228 Hamburger, H., 219, 229 Hammersley. J. M.. 210, 229 Hardy, G. H., 10. 220,229 Hausdorff, F., 219,229 Hille, E., 206. 229 Hoffman, K., 204,206, 216, 220,229 Holland, A. S. B., 210, 212, 229 Holub, J. R., 209. 210,232 HruSEev, S. V., 223, 22Y

I Ingham. A. E.. 220,221,229

Katsnelson, V. 8.. 221, 223, 229 Knopp. K., 21 I , 229 Kothe, G., 208, 229 Koosis, P.,139, 218,229 Korevaar, J., 213,229,231 Korobeinik, Ju. F., 221, 222,226,227,229 Krein, M. G., 46. 209,219,225,228, 229, 230 Krotov, V. G., 204,230 Kwapien, S., 204. 230

L Landau, H. J., 218,230 Lerer, L. E., 209, 230 Levin, B. Ja., 210, 219-222, 230 Levinson, N., 90, 1 I I , 143,210,213-218,224. 230

Lin, B., 209, 230 Lindelof, E., 54, 212, 231 Lindenstrauss, J.. 203, 205, 207, 208,230 Littlewood, J. E.. 158, 220, 229 Liusternik, L. A , , 209, 229 Lorch, E. R., 208.230 Lorentz, G. G., 205, 230 Luxemburg, W. A. .I.213, , 230, 23/ Lyubarskii, J. I., 220, 222,230 M Malliavin. P., 142, 218, 226 Marcinkiewicz, J.. 204, 231 Markushevich, A. I., 44, 210,231 Marti, J. T., 203, 204, 23/ Martirosjan, V. M., 222,231 Meletidi, M. A,, 206, 209, 228, 231 Milman, D. P., 209.230 Morris, P., 205, 227 Muntz, C. H., 205.213.231

J James, R. C., 206, 207,229 Jensen. J. L. W. V., 21 I , 229 Johnson, W. B., 207,229

K Kac, M., 221.22Y Kadec. M. I., 210, 223,229 Kahane. J. P., 139, 218, 229 Karlin, S., 203, 204, 207, 229 Karnick, H., 210. 233

N Natanson. I. P.,219,231 Nehari, Z.. 207, ,731 Neuwirth, J. H.. 209,228 Neville, C. W.. 220. 231 Nudelman, A . A . 219.230 0 O'Hara, P. J., 213. 231 Orlicz, W., 208. 23/

239

Author Index Oskolkov, V., 222. 231 Ostrovskii, I. V., 230 P Paley, R. E. A. C., 38, 42, I I I , 139. 206, 209, 213-217, 219-221, 223. 231 Pavlov. B. S.. 223. 231 Pelczynski. A.. 203 205, 208. 226, 230. 231 Peterson, D. R.. 218. 231 Phragmen, E., 212. 231 Plancherel, M., 94. 212, 214, 231 Pollard, H., 207, 215, 226,231 Pdya. G., 94, 138. 158, 212-214. 216. 220, 229, 231 Povzner, A,, 232

R Rahman, Q. I . , 213. 228 Redheffer, R. M., 212, 215-218.225.232 Retherford, J. R.. 209, 210, 232 Riesz, F., 34.48. 206, 210, 220, 228, 232 Rogers, C. A.. 204.228 Rosenbdum, J. T.. 220,232 Rosenthal, H. P.. 207, 229 Rota, G. C.. 210. 226 Royden. H. L.. 205. 232 Ruckle, W. H., 209. 232 Rudin, W.,94,97,205, 213-215,232 Russell, D. L.. 210, 232 Rutman, M. A., 209,230 S

Schaeffer. A. C . , 2 I 2 2 14. 222 224, 227 Schiifke, F. W.. 210, 232 Schauder, J., I , 206. 232 Schmeisser, G . . 2 13, 228 Schonefeld. S., 204, 232 Schwartz, J. T., 206, 219, 220. 227 Schwartz. L., 139. 213. 215, 217, 218. 232 Sedleckii. A. M.. 217, 218, 220 224,232,233 Sengupta. A,, 210. 233 Shah, S. M., 21 I . 233 Shannon, C. E.. 107, 233 Shapiro, H. S.. 160. 206, 2 13, 220, 233 Shields, A. L.. 160. 220. 233 Shisha. 0..215, 231 Shohat, J. A,. 219, 233

Simon, P., 204, 208,227 Singer. I . , 203,205.208.209.221,230,23/, 233 Sjolin. P., 204, 208, 227 Stepanoff, W., 206,233 Stieltjes, T. J., 219, 233 Stray, A., 222, 233 Svedenko. S. V., 220,233 Sz.-Nagy, B., 34, 48, 206, 210, 232 SzBsz, O., 201, 213, 216, 233 Szego, G., 16,206,212,213,231,233

T Talaljan, F. A., 233 Tarnarkin, J. D., 219,233 Taylor, A. E., 36, 186, 233 Taylor, B. A,, 220, 233 Titchmarsh, E. C., 75, 83, 206, 210-212, 214. 220. 233 Turku. H., 220, 233 Tzdfriri, L., 203, 207, 208. 230 V

Valiron, G . . 210. 233 Verblunsky, S., 224, 233 Vitali, G.. 206, 234

W Wallen. L. J., 210, 234 Walsh, J. L., 205, 207, 223, 234 Weaver, W., 107, 233 Weierstrass, K., 54, 205, 21 I , 234 Whittaker, E. T.. 107. 234 Whittaker, J. M.. 215, 234 Wiener, N., 38. 42, I 1 I. 139, 206, 209. 213 217. 219- 221, 223,231 Williams, D. L.. 220,228, 233 Wolibner, W., 205. 234

Y Young, R. M., 215. 218-223. 2.34 Z Zaharjuta, V. P., 222, 227 Zippin, M . . 207, 208, 229. 230 Zygmund. A., 206.234

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A

Absolute basis, 3 Absolutely convergent series, 2, 204 Almost periodic exponential sum, 84 Almost periodic function. 14. 206 Approximation property, 26, 205, 207

B Banach space, 1 Banach- Steinhaus theorem, 30 Bari basis, 46, 5 I Bari’s theorem, 45. 210 Basis, I , 203 for A’, I I absolute, 3 Bari, 46, 5 1 Bessel, 36, I53 bounded, 3 I for C[a,b],3, 204 conditional, 204, 208 for the dual space. 27-29, 207 of eigenfunctions, 47 equivalent bases, 30, 208 for H2. 10 Hamel, 1 Hilbert, 37, 153, 167 for lp, 2 for LpI - n,n],203 for Lp[O.I],203 normal. 26. 207 orthonormal, see Orthonormal basis for P, 106 perturbation of. .see Stability Riesz, w e Riesz basis Schauder, I , 207

stability of, see Stability unconditional, 3, 13, 204 uniqueness of, 208 Basis problem, 2 Bergman kernel, 18,207 Bernstein’s inequality, 84, 86, 104, 141,212 Bernstein polynomials, 3, 5 , 205 Bessel basis, 36, 153 Bessel functions, 68 Bessel sequence, 146, 153-161, 180, 220 of complex exponentials, 162 of powers, 157 Bessel’s inequality, 7 Biorthogonal sequences, 28,29, 31.32, 36,37, 41, 51, 148, 224 Birkhoff-Rota theorem, 46, 210 Boas-Pollard theorem, 22, 207 Borel’s theorem, 69, 21 1 Borel-Caratheodory inequality. 75 Boundary function, 174 Boundary-value problem, 46 Bounded basis. 31 Bounded from below, 36, 157, 186 Bounded variation, function of, 1 I 1 Bounds for a Bessel sequence, 154 for a frame, 185 for a Riesz-Fischer sequence, 154 C

Canonical factorization, 65 Canonical product, 56. 69-74 of genus zero, 70, 7 I lower bound for, 73.21 I Cardinal series, 107, 109, 151, 215 Carleman’s formula, 87-93, 115, 117, 213 241

242

Subject Index

Carleman's theorem, I14 Closed graph theorem, 35, 169 Closed linear span, 27 Closed sequence, see Complete sequence Coefficient functionals, 22-28.40, 207 Compact operator, 40,41,45 Complete interpolating sequence, 171, 175 Completeness radius, 137-143, 166 Complete orthonormal sequence, see Orthonormal basis Complete sequence, 6, 13, 19-22, 207 ofcomplexexponentials, 111-143,215-219 of powers. 21 Conditional basis, 204, 208 Corona theorem, 160

D Dalzell's theorem, 15, 206 Deficiency, 113, 116, 122, 143, 215 Density, 138, 139, 143 P6lya maximum density, I38 upper, 138 of zeros, 90,213 Density zero, I39 Derivative, of an entire function inequalities for, 84. 86, 99, 104 order and exponential type of, 67, 74 zeros of, 79 Dimension linear, 2 orthogonal, 13 Dirichlet kernel, 199 Disk algebra, 26, 204 Dual space, 20,27-29 basis for, 27-29, 207 Duffin-Eachus theorem, 44,50,210 Duffin-Schaeffer theorem, 190,222 Dvoretzky- Rogers theorem, 204

E Eigenfunction expansion, 46,47 Entire function of exponential type, see Exponential type, function of factorization theorems, 54-58, 74-79 of finite order, 62-68 interpolation by, 58, 106, 107, 151

uniqueness of, see Uniqueness, of an entire function zeros of, see Zeros, of an entire function Equiconvergent, 197, 223,224 Equisummable, 201 Equivalent bases, 30, 208 Equivalent inner products, 32 Equivalent sequences, 167- 170 Exact frame, 186, 188, 196 Exact sequence, 1 12, 126, 149, 2 I5 Excess, 113, 137, 143,215-218 Exponent of convergence, 66 connection with order, 66 in terms of n(r), 67 Exponential type, 71 of a derivative, 74 in terms of Taylor coefficients, 74 Exponential type, function of. 53, 61 almost periodic, 84 bounded on a line, 80-90,212-214 bounded on a sequence, 212 density of zeros, 90, 213 inequalities for, 82,84,94,99, 104, 107, 108, 212,213 interpolation by, 106-108, 151, see also Interpolation Lz on the real axis, 100-109 Lpon the real axis, 93-100, 104,214 representation by Fourier integrals. 101, I03 uniqueness of, 90-92, 105, 107. 108, I I I , 216

F Factorization theorems Hadamard's, 74.21 I Weierstrass's, 54, 55, 21 1 Fejer-Riesz inequality, 161, 220 Finite interpolation, 147 Finite moment problem, 147 Finite order, 63 Finite rank operator, 26. 205 Fourier coefficients, 6, 9 Fourier inversion formula, 101, 150 Fourier series expansion, 6,9, 197 mean convergence, 9 pointwise convergence, 9 Fourier transform, 9, 101, 127, 142 inverse, 150

Subject Index

243

Fourier- Stieltjes integral. 83, 172 Fourier- Stieltjes transform, 53 Frame, 184- 196. 222, 223 bounds for, I85 stability of, 191 196 Franklin system, 204 Fredholm alternative. 40, 45 Free, see Minimal Functional Hilbert space, 16-19 interpolation in. I58 Fundamental sequence. .we Complete sequence G Gamma function, 76 G a p theorem, for an entire function, 105 Genus, of a canonical product, 56 Genus zero, 71 Gram determinant. 147 generalized, 152 Gram matrix, 32, 157 Gram- Schmidt orthogonalization procedure, 15, 204 Gurarii-Gurarii theorem, 209 Gurarii- Meletidi theorem, 209

Infinite product, 54.21 1 Integrability on a line, 93- 100 lnterpolatingsequence, 171,175,180, 183, 191 Interpolation, 145 in B,, 161 duality with basis theory, 169-171, 222 in E:, 178 finite, 147 in a functional Hilbert space, 158 in H2,158 in P. 170 184,221 solution of minimum norm, 147, 152 trigonometric, see Trigonometric moment problem

J Jensen’s formula, 59-62, 119, 124.21 I

K Kadec’s 4-theorem, 42, 122, 131, 196,210 Karlin’s theorem on biorthogonal bases, 29, 207 on normal bases, 26, 207 Kernel function, see Reproducing kernel Krein- Milman-Rutman theorem, 39, 209

H

Haar system. 203 Hadamard’s factorization theorem. 74, 21 1 Hadamard’s inequality, 66. 69 Hahn-Banach theorem, 20 Hamburger moment problem, 219 Hamel basis. 1 Hausdorff maximal principle, 13 Hausdorff moment problem, 219 Hermitian matrix. I59 Hilbert basis. 37. 153, 167 Hilbert matrix. 158 norm of, 161 Hilbert space, 6. 206 Hilbert-Schmidt operator. 45, 51 Hilbert’s inequality. 161. 220 Hurwitz’s theorem. 173

I Independent, scc Minimal Infinite deficiency. 143, 165

1 Lagrange interpolation series, 150- I5 I , 176, 219 Laguerre’s theorem, 79 Landau’s theorem, 218 Legendre polynomials, 15 Legendre’s duplication formula, 79 Levinson’s theorem on completeness of exponentials in C [ - n D , d ] , 117,216 on completeness of exponentials in L ” [ - ~ I , A 118, ] , 216 on density and the completeness radius, 138, 218 on equiconvergence of ordinary and nonharmonic Fourier series, 224 on exponentials close to the trigonometric system. 119, 122,216 1.i.m.. see Limit in mean Limit in mean, 199

244

Subject Index

Linear operator, see Operator Linked. 130

M Maximum modulus function, 62 Maximum principle, 62,80,95,96, 174 Mean-value formula, for analytic functions, 17 for harmonic functions, 59 Measurable sequence, I38 Minimal, 28, 130 Minimum modulus, 21 1, 212 Mittag-Leffler function, 68 Moment, 21, 146,219 Moment problem, 146-153,219 relation to bases. 169-171, 222 Moment sequence, 146, 186 Moment space, 146, 154, 169 and equivalent sequences, 167-170 of a Riesz basis, 148 Muckenhoupt condition, 223 Miintz’s theorem, 91, 93, 213 N

Natural basis, 2, 3, 7 Nonharmonic Fourier frame, 185 Nonharmonic Fourier series, 42, 209, 221 equiconvergence with ordinary Fourier series, 197-201 pointwise convergence, 197-201 Normal basis, 26. 207

0 w-independent, 40,46 Onto mapping, 182 Open mapping theorem, 30, 188 Operator bounded from below, 36, 157, 186 compact, 40,41,45 of finite rank, 41 Hilbert-Schmidt, 45.51 positive. 34, 48 square root of, 34 Operator of finite rank, 41 Order, 62-68 of a canonical product, 69 of a derivative, 67 formula for, 64

of a product, 67 in terms ot Taylor coefficients, 67 Orlicz’s theorem, 208 Orthogonal, 13,206 Orthogonal polynomials, 206 Orthogonal system, 26 Orthonormal basis, 6 for A’, I 1 for H2, 10 for P , 7 for L’[O,n], 14 for L2[-n,n].7 for P,106 stability of, see Stability Orthonormal sequence, 14, 15

P Paley-Wiener space, 19, 105-109, 112, 126, 149, 215 basis for, 106 interpolation in, 170-184 Paley-Wiener stability criterion, 38, 42, 48, 209 Paley-Wiener theorem on finite Fourier transforms, 100-105, 214 on minimal sets of complex exponentials, 130, 217 on stability of bases, 38,41, 209 Parseval’s identity, 6, 9, 12, 106 generalized, 7 Partial sum operator, 24 adjoint of, 27 Pelczynski-Singer theorem, 208 Perturbation of a basis, see Stability Phragmen-Lindelof method, 80-87 Phragmen-Lindelof theorem, 80,212 Plancherel’s theorem, 100, 106, 195 Plancherel-Polya theorem on differentiation in Er, 99, 214 on an inequality for functions in E:, 93, 214 Point-evaluation functional, I6 Pointwise convergence of Fourier series, 9, 197, 201 of nonharmonic Fourier series, 197-201 Poisson summation formula, 105 Polar decomposition, 48 Polya maximum density, 138, 143 Positive matrix, 19 Positive operator, 34.48

Subject Index

245

Powers, completeness of, 2 I . 22 Primary factors, 55 Property SAIN, 205

Q Quadratic form. 157, I59 Quadratically close, 45,46, 5 I

R Redheffer’s theorem, 137. 21 8 Reflexive Banach space. 28,204 Reproducing kernel, IS 19, 206 Bergman kernel, 18 for P, 107 Szego kernel, 16. 158 Riemann Lebesgue lemma, 200 Riesz basis, 30-37, 169. 188. 208. 221 of complex exponentials. 42, 148. 151. 170177, 181, 191, 192. 196- 201 Riesz (F. and M.) theorem, 215 Riesz-Fischer sequence, 146, 153 161. 180. 220 of complex exponentials. 162- 166 Riesz- Fischer theorem, 7 Riesz representation theorem, 20, 53, 91, 152

S Sampling theorem, 107 Schafke’s theorem, SO, 210 Schauder basis, 1, 207 Schauder’s system, 3. 204 Schauder’s theorem. 4 Schur’s lemma, 159 Sedleckii’s theorem on equiconvergence of ordinary and nonharmonic Fourier series, 223 on Fourier Stieltjes transforms, 177, 222 Self-adjoint operator, 186. 189 formula for norm, 48 Separable. 2, 6 Separated sequence. 98, 179-181, 184, 221 Sequence Bessel, .see Bessel sequence closed, sec, Complete sequence complete, see Complete sequence density of, see Density equivalent sequences, 167, I70

free, 28 fundamental, see Complete sequence independent, 28 interpolating. see Interpolating sequence measurable, I38 minimal, 28, 130 w-independent, 40.46 quadratically close sequences, 45, 46, 5 I Riesz- Fischer, see Riesz-Fischer sequence separated, 98, 179-181, 184, 221 symmetric. 139, 148, 151 total, see Complete sequence uniform density 1, 212, 214 uniformly minimal, 160 Set of uniqueness, 112, 174 Shannon’s sampling theorem, 107 Sine type, function of, 171-174, 224 zeros of, 173, 174 Square root, of an operator, 34 Stability of bases in Banach spaces, 37-41,209 of complete sequences in Banach spaces, 209 of complete sequences of complex exponentials, 131-137, 217, 218 of frames, 191-196 of interpolating sequences, 179- 184 of nonharmonic Fourier series, 190-197 of orthonormal bases in Hilbert space, 41 51 of Riesz bases of complex exponentials, 190 197 of the trigonometric system, 42-44, 118122, 216 Sturm-Liouville series, 47. 210 Sturm-Liouville system, 46 Subharmonic function, 94, 95 Summability, 201 Symmetric sequence, 139, 148, 151 Szego kernel, 16, 158

T Total sequence, see Complete sequence Trigonometric moment problem, 145, 170, 219 generalized, 148 Trigonometric polynomial approximation by. 8 inequality for, 86

246

Subject Index

Trigonometric system, 7, 8, 20, 1 12- I 17, 203 completeness of, 8, 20 stability of, 42-44, 118- 122, 196, 216, 218, 223,224 Type, 71 U

Unconditional basis, 3, 13, 204 Unconditionally convergent series, 2, 204 Uniform approximation, 3, 129 Uniformly minimal, 160 Uniqueness, of an entire function, 90-92, 105, 107, 108. 1 11, 216 Unitary operator, 48 Upper density, 138, 143

V Vitali’s theorem, 14, 206

W Wallis’s product, 58 Weak basis, 203 Weak convergence, I36 Weierstrass primary factors, 55 Weierstrass’s approximation theorem, 3, 8, 21, 113, 205 generalizations of, 205 Weierstrass’s factorization theorem, 54, 55, 21 I Weighted interpolation, 158 Wolibner’s theorem, 205 2

Zeros, of an entire function density of, 90,213 functions of nonintegral order, 75, 76 and growth, 59-62,64-66,69-73,75,85 real, 79

93