J Integral Equations and Operator Theory ON THE NEVANL INNA-P ICK INTERPOLAT ION PROBLEM FOR GENERAL IZED ST IELT JES FUNCTIONS

Dedicated to the memory of M. G. Kre~n The solutions of the Nevanlinna-Pick interpolation problem for generalized Stielt-jes matrix functions are parametrized via a fractional linear transformation ver a subset of the class of classical Stieltjes functions. The fractional linear trans-formation of some of these functions may have a pole in one or more of the interpolation points, hence not all Stieltjes functions can serve as a parameter. The set of excluded parameters is characterized in terms of the two related Pick matrices. 1 In t roduct ion The objective of this paper is to study the Nevanlinna-Pick interpolation problem in the class of generalized Stieltjes matrix functions. The Nevanlinna-Pick problem for classical Stieltjes functions was considered by M.G. KreYn and A.A. Nudelman [8] for the scalar case; the matrix case appeared in [6] and its tangential and two-sided generalizations in [1], [3]. In these problems the two kernels in (1.2) below are nonnegative. Scalar generalized Stieltjes functions have been treated by M.G. Kre~n and H. Langer in [7] under the assumption that the first kernel in (1.2) below has a finite number of negative squares and the second one is nonnegative. In this paper we consider the matrix case and assume that both kernels have a finite number of negative squares. V. Derkach in [4] studied the moment problem in the same generality. Throughout this note we work with meromorphic matrix functions, and when S is such a function, we denote the set of points of analyticity by p(S). Definit ion 1.1 An m x m matrix function S belongs to the generalized Stieltjes class $ ~ if it is meromorhic i