Multiple interpolation in the half-plane in the class of analytic functions of finite order

Abstract

The problem of multiple interpolation in the class of entire functions of finite order was solved in [1]. The analogous problem for the class of functions of finite order in the upper half-plane C + = {z: Imz > 0} was discussed in [2], where necessary and sufficient conditions for its solvability were found. The difficulties that appear upon passage from the interpolation problem in classes of entire functions to classes of functions that are analytic in the half-plane C + are caused by the possibility of interpolation nodes accumulating at points of the real axis. The sufficient conditions found in [3] eliminate the possibility of such accumulation and make it possible to apply the methods and theory of entire functions. In the present paper we consider a problem more general than that of [2], eliminate the above-noted shortcomings, and find necessary and sufficient conditions for solvability of our problem. In its formulation, our problem is close to problems on free interpolation in H ~176 , since the values of the derivative of a function at interpolation nodes are subjected to some natural constraints. We will use ideas from [1, 2], as well as ideas due to Jones [3] for solution of problems on free interpolation in H ~176 Let Lo, ~]+ denote the class of functions that are analytic in C + and of order _ 1. Definition. A divisor D = {an, qn} (i.e., a set of different complex numbers an E C +, n = 1, 2,..., whose limit points all lie on the rea/axis, together with their multiplicities qn, qn >_ 1, is an integer) is said to be an interpolation divisor in the class Lo, oo] + if, for any sequence of numbers {bnk}, k = 1,..., q,, n = 1, 2,..., that satisfies the conditions (Irna,)~-~lb~kl In + In maxl<k<a~ (k-l)! lim