Solution of a multiple Nevanlinna–Pick problem for Carathéodory functions via orthogonal rational functions in the matrix case

Abstract

We consider an interpolation problem of Nevanlinna–Pick type for matrix-valued Carathéodory functions, where the values of the functions and its derivatives up to certain orders are given at finitely many points of the open unit disk. For the non-degenerate case, i.e., in the particular situation that a specific block matrix (which is formed by the given data in the problem) is positive Hermitian, the solution set of this problem is described in terms of orthogonal rational matrix-valued functions. These rational matrix functions play here a similar role as Szegő's orthogonal polynomials on the unit circle in the classical case of the trigonometric moment problem. In particular, we present and use a connection between Szegő and Schur parameters for orthogonal rational matrix-valued functions which in the primary situation of orthogonal polynomials was found by Geronimus. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)