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

{