The representation of functions by exponential series

CHAPTER 11

INTERPOLATION In this introduction, from the various aspects of the theory of interpolation by analytic functions, we discuss only one: the free interpolation (the Carleson interpolation) which gravitates to the theory of the distribution of the values of analytic functions and whose signs are "the ideal spaces of the given interpolation" (a space of functions X is said to be ideal if ~ e X , I~I~I~I ~ X ). Of course, the mentioned "free" or "ideal property" need not appear as simple as shown in the parentheses, especially when one considers interpolation by germs (of analytic functions) of unbounded height, or interpolation by functions which are smooth up to the boundary of the domain, etc. Four of the five problems of Chap. 11 (1.11-3.11, 5.11) are devoted to free interpolation and we hope that these lines will not cause the reader any divergence regarding the interpretation of the term. The free interpolation is the topic (fundamental or peripheral) of other sections of the present Collection (7.4, 2.9, 2.12) but, nevertheless, the presented material does not give a complete representation of the present development of this subject; for further information we refer the reader to the survey of S. A. Vinogradov and V. P. Khavin [Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR, 47, 15-54 (1974); 56, 12-58 (1976)]. Finally, in the last 15 years, the simple but important relation between the theory between the theory of interpolation (or, differently, the theory of moments) and the study and classification of biorthogonal expansions (bases) has been completely elucidated. Without touching upon the continual analogues of these correlations, interesting for the spectral theory, we mention only that each pair of biorthogonal families are vectors of the space X, xl are functionals,

~={$~,

=[ X~Xr

(x%

and biorthogonality means that <x%, x~> =

6%~) generates an interpolation problem regarding the description of the space l

]X~]~ ~

l

{<$'~>]Xc~) of the coefficients of the formal Fourier series the property "free" in this interpolation

~

~<~,~>~X"

(i.e., the property of the space

As a rule, the ~

of being

ideal) is equivalent to the unconditional basis property of the family ~ in the closure of its linear span. This circumstance, used in an explicit form already by T. Carleman and S. Banach, now plays an important role in the interaction of the interpolational methods with the spectral theory, since this latter turns out to be a fundamental supplier of biorthogonal families, interesting both for themselves and for the theory of functions. Such families are usually the families of eigenvectors (or root vectors) of some operator (in the theory of functions, either differentiation or its isomorphic operator of the inverse shift): m ~ = ~,

~.

Therefore,

the properties of the equation

fined on the set o) and the properties of the space

~=~ ~X

(~

is a given function, de-

depend on the supply of multipliers

of the family ~. (By the multiplier of a family ~ we mean here a linear operator, acting in X and mapping the vector x% into the vector ~ ( ~ X , ~ G ~ , where ~ is a function defined on o; the multiplier itself is also denoted by the letter V.) These multipliers ~ may be functions of the operator T ( ~ =~CT)) and then there arises another, namely the multiplier problem of interpolation (the search of ~ for a given ~), whose solution leads frequently also to the solution of the initial moment problem ~=~. Bases and interpolation occur (unfortunately, in an implicit manner) in problems 1.11, 3.11, 7.4, 2.12, but (just as in the previous paragraph) we cannot go here into more details or into the enumeration of the literature and we refer the reader to the recent survey of N. K. Nikol'skii [Tr. Mat. Inst. Akad. Nauk SSSR, 130, 50-123 (1978)]. We only mention that the multipliers from 3.11, by D. Sarason's known theorem, are related to the mentioned multipliers of the biorthogonal families. Interpolation problems arise also at the attempts of the localization of ideals in the theory of functions which are periodic in the mean, where they are the dual forms of the problems of the expansion of functions into the Weierstrass canonical product. This topic is

2275

reflected in the problems of Sec. obtained from the references.

1.11, and numerous publications

on related problems can be

The multidimensional interpolation by analytic functions is almost not touched upon at all in this chapter. The known results in this topic carry either a classical character and are presented in textbooks (the problems of the continuation of holomorphic functions from submanifolds) or are connected with free interpolation on discrete subsets of the corresponding domains of the space

~;

regarding this, see, e.g., E. Amar, "Suites d ' i n t e r p o l a t i o n p o u r

C

les classes de Bergman de la boule et du polydisque de , " Univ. Paris XI, Orsay (1977)~ preprint No. 224, which also indicates other papers on the same subject. On the other hand, in Sec. 5.11 the free interpolation on nondiscrete subsets of a polydisc is considered.

2276

1.11.

THE REPRESENTATION OF FUNCTIONS BY EXPONENTIAL

I.

Let L be an entire function of exponential

SERIES*

type, let k = k L be its zero divisors

[k(X) is the multiplicity of a zero of the function L at the point X, ~ ~ Borel associated function of L, i.e.,

b~=~.~l~(~)~•

(~),

], let y be the

where the closed contour C

includes the closed set D containing all the singularities of the function y. There exists a family of functions {@k,%:0 < k < k(%)}, regular in ~,D and biorthogonal with the family {zSeXZ:0 ~ s < k(%)} in the sense that I

where ~ a ~ is the Kronecker delta (the construction of Sk,% can be found in [I, p. 228]). each function f, analytic in D, one can associate its Fourier series

~=0

To

(I)

C

The following uniqueness theorem is known ([I, p. 225]): If the set of zeros of the function L is infinite and D is a convex set, then ak,% z 0 ~ f ~ 0. In the available proof one makes use in an essential manner of the convexity of the set D. Problem I. vexity of D? 2.

Is the uniqueness

theorem valid without the assumption regarding the con-

Let D be the smallest closed convex set containing all the singularities of the

function ~ and assume that the function L has only simple zeros [i.e., ~(~)~4 , ~ G ~ ]. In order that the series (I) should converge to f in the interior i n t D of the set D, it is necessary and sufficient to have the following conditions: ~{~)>0, i~i~D(~,

a) l~(~)l>e [~(~-g]LM ~ @ = ~ ,

6>0 is arbitrary; b) there exist numbers p, p > 0, and

D~ , O c ~ t o o , such

that tLCX)I>eP~INI=~,~ . Condition a) ensures the convergence of the series (I) in the domain intD, while condition b) ensures that the sum of the series coincides with f. Problem 2.

Is condition b) a consequence of condition a)?

A negative answer would mean that there exists a series to its generating function. 3.

If i n t D

is an unbounded convex domain, ~ o 0 , O ] c ' ~ D

the support line of the domain intD, then The angle

~

may coincide with

the corresponding

line

~

.

• Let

L~

. 0

, and if ~ : ~ 0 0 5 ~ + ~ $ ~ - K ~ ) = 0

~ varies between the limits

if the boundary ~(@)=K~

(I) which is convergent but not

-~4~(04~)

is 9

~D, starting with some place, goes along

and %~(q)

(2)

and in the above-mentioned cases we may also have @ = • All the zeros of the function L are assumed to be simple. Let {~%:k(%) > O} be a family biorthogonal with {e%Z:k(%) > 0}, and let

*A. F. LEONT'EV. USSR.

Bashkir Branch, Academy of Sciences of the USSR, ul. Tukaeva 50, Ufa 450057,

2277

L(.Oe t

-

L'~) 3 t-~

'

'

0

Because of the conditions (2) the functions ~X are regular outside D, continuous up to the boundary and bounded (by constants depending, in general, on the roots X). Let B(D) be the class of all functions and such that

~(~)=0Q~[~), ~ > I

Fourier series venient

,~D

f that are analytic

,~-~oo

(I), assuming C = ~D [and

To each function

qJ~.X=~,~(~)>0

to renumber the zeros of the function

in intD, ~ ,~[D)

continuous

in D,

, we associate

the

] and in this section it is con-

l. :i~} =[~:~(~)>01.

The problem consists in the convergence of the series (I) in intD to the function f. Assume that L satisfies additionally also the following condition: There exists a system of closed contours ~8~=0

~(~4)

and a system of curvilinear

,[~D~=00(~-~l~l)

, containing

rings

~ = t ~ r ~ : l % - ~ l ~ e gLz~] ,

6~>0 ,

these contours, having the properties:

a) for any y, y > 0, and E, e. > 0, one has

~,i,n~ ~,F~ lq CX)= +oo ,where H CX)= ~o0~ILCX)I.

The function H for k > K(y, s) is larger than larger

than

the angle

~(-~0+1~)-6 I~1<~ o +~ ,

are the parts of

~

on

~

, where

Fk i s

is that part of

~

~(~)-g

the

part

~ ,'

on of

r k which

~O "O

lying in the small angles

I~-~oI.~ ,

I@+~I~,

~(~0)-6

in the

on

ff

complement

]h~]~-~-~',~ '

and

, of

~

respectively;

the boundary of the curvilinear

"~ (where we assume that Ck lies inside ~k), then the lengths of the

curves ?~, C~, and C~ as k § ~ are exp c) in the ring between the zero Xk. E C

lies

which lies in the angle

b) if Ck and Ck are those parts into which Fk partitions semiring

larger than

[O(1)rk] ;

Fk and Fk+ ~ there lies only one zero of the function L, namely

Under the mentioned conditions, intD, then

it is shown

[2] that if

~eS(D)

and E is a compactum,

~V

Ave l<e so that I : % - ~ X I~ V~4

case

-~)(.D),

I

~

~-~O.

,~o=~oCE)>O,~,% (zeE),

In [2, 3] it is shown how one can reduce the general case to the

V

when from f one requires only analyticity

Problem 3. Show that for any domain (I), a), b), c).

intD there exists a function L with the properties

LITERATURE I ~

2.

3.

2278

in intD.

CITED

A. F. Leont'ev, Exponential Series [in Russian], Nauka, Moscow (1976). A. F. Leont'ev, "On the question of the representation of analytic functions in an infinite convex domain by Dirichlet series," Dokl. Akad. Nauk SSSR, 225, No. 5, 1013-1015 (1975). A. F. Leont'ev, "On a certain representation of an analytic function in an infinite convex domain," Anal. Math., 2, No. 2, 125-148 (1976).