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Published online by Cambridge University Press: 25 June 2025
A basic interpolation problem, which includes bitangential matrix versions of a number of classical interpolation problems, is formulated and solved. Particular attention is placed on the development of the problem in a natural way and upon the fundamental role played by a special class of reproducing kernel Hubert spaces of vector-valued meromorphic functions that originate in the work of L. de Branges. Necessary and sufficient conditions for the existence of a solution to this problem, and a parametrization of the set of all solutions to this problem when these conditions are met, are presented. Some comparisons with the methods of Katsnelson, Kheifets, and Yuditskii are made. The presentation is largely self-contained and expository.
1. Introduction
This paper presents a largely self-contained expository introduction to a number of problems in interpolation theory for matrix-valued functions, including the classical problems of Schur, Nevanlinna-Pick (NP), and Caratheodory-Fejer (CF) as special cases. The development will use little more than the elementary properties of vector-valued Hardy spaces of exponent 2.
Moreover, by exercising a little care in the choice of notation, most of the analysis for all three of the classical choices of Ω+ mentioned above can be carried out in one stroke. Table 1 serves as a dictionary for the meaning of the symbol that is appropriate for the region Ω+ in use. In order to describe the BIP we need to introduce some notation.
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