On the reproducing kernel of a Pontryagin space of vector valued polynomials

TL;DR

This paper characterizes when the reproducing kernel of a Pontryagin space of vector polynomials can be described by a generalized Nevanlinna pair of matrix polynomials, providing a clear criterion for such spaces.

Contribution

It establishes necessary and sufficient conditions linking the reproducing kernel of Pontryagin spaces of vector polynomials to generalized Nevanlinna pairs of matrix polynomials.

Findings

Characterization of reproducing kernels via Nevanlinna pairs

Necessary and sufficient conditions for kernel determination

Extension of kernel theory to vector-valued polynomial spaces

Abstract

We give necessary and sufficient conditions under which the reproducing kernel of a Pontryagin space of $d \times 1$ vector polynomials is determined by a generalized Nevanlinna pair of $d \times d$ matrix polynomials.