ABSTRACT
The notion of free interpolation passed through several stages during the last three decades.- Starting about 100 years ago as a Hadamard theorem on the existence of an entire function/ with arbitrary (prescribed) valuesf(zn) =an on an unbounded sequence of complex points Zn E C, limn lznl = oo, it became an important tool for different purposes (e.g., for exponential-polynomial description of solutions of convolution equations, for studies of bases of rational functions and exponentials, for discovering the multiplicative structure of holomorphic functions, and so on). For example, free interpolation is very useful in studying Hardy classes HP, and especially, the algebra H 00 of all bounded analytic functions on the unit disc D = {z E C: lzl < 1}. For H 00 , the evolution of the free interpolation notion includes: The well-known study of subsets a c 0 for which the restrictions space H 00 I a is
(C)
tion). General multiple free interpolation, and free interpolation with respect to an arbi-
trary sequence { 8n}n>t of inner function where the problem is to find a function/ E HCXJ with prescribed projections onto the correspondingz*-invariant
278 Nikolskii and Volberg
Finally, the most general free interpolation with respect to an arbitrary sequence {8n}n>l of H 00 functions; this last problem can be described and solved as follows.
