The Nevanlinna-Pick interpolation problems and power moment problems for matrix-valued functions. III: The infinitely many data case
The paper deals with the multiple Nevanlinna-Pick (NP) matrix interpolation problem with a denumerable set of nodes in the Nevanlinna class \(\mathcal{N}_p\) \((p \geq 1)\) of \( \mathbb{C}^{p \times p}\)-valued functions \(f(z)\) which are holomorphic in the open upper half plane \(\pi^+\) (NP\((\mathcal{N}_p)_\infty\) problem) and the full Hamburger matrix moment problem (HM\((\mathbb{R})_\infty\) problem). The close connection between an NP\((\mathcal{N}_p)_\infty\) problem and HM\((\mathbb{R})_\infty\) problem associated with the block Hankel vector of the former is studied. This connection allows to reduce the solution of the NP\((\mathcal{N}_p)_\infty\) problem to the study of a certain HM\((\mathbb{R})_\infty\) problem. The case of infinitely many nodes is handled by means of analysis of the limiting behaviour of approximating problems involving only finitely many nodes. The notion of the block Hankel vector of an NP\((\mathcal{N}_p)_\infty\) problem is first introduced and an integral and explicit correspondence between solutions to a solvable NP\((\mathcal{N}_p)_\infty\) problem and solutions to the associated HM\((\mathbb{R})_\infty\) problem with that block Hankel vector is established. Then a congruence relation between the infinite generalized block Loewner matrix \(L_\infty\) of NP\((\mathcal{N}_p)_\infty\) problem and the infinite block Hankel matrix \(H_\infty\) built on the block Hankel vector of the former problem is made, which is a generalization of the classical Loewner-Hankel matrix relation. This link allows to obtain a parametrization of the solutions to the multiple NP\((\mathcal{N}_p)_\infty\) problem in both nondegenerate and degenerate cases by a linear fractional transformation from the corresponding results of the HM\((\mathbb{R})_\infty\) problem. The transformation is based on the Schur algorithm involving matrix continued fractions as well as orthogonal polynomial matrices simultaneously.
infinite generalized block Loewner matrix
scientific article; zbMATH DE number 1421385
scientific article