A Basic Interpolation Problem

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 Hilbert 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.