THE GENERALIZED BITANGENT CARATHÉODORY-NEVANLINNA-PICK PROBLEM, AND -INNER MATRIX-VALUED FUNCTIONS

Abstract

This paper is a study of the problem of describing holomorphic matrix-valued functions on the unit disk with (the Carathéodory class ) such that , where , , and are particular matrix-valued functions with and inner and in , and is the Smirnov class of matrix-valued functions of bounded type on . The matrix extrapolation problems of Carathéodory, Nevanlinna-Pick, and M. G. Krein reduce to this problem for special and , as do even the tangent and -tangent problems when there is extrapolation data for and not on the whole Euclidean space but only on chains of its subspaces. In the completely indeterminate case the solution set of the problem is obtained as the image of the class of holomorphic contractive matrix-valued functions on under a linear fractional transformation with -inner matrix-valued function of coefficients on . The arising in this way form a class of regular -inner matrix-valued functions whose singularities appear to be determined by the singularities of and . The general results are applied to Krein's problems of extension of helical and positive-definite matrix-valued functions from a closed interval.