Kernel decompositions for Schur functions on the polydisk

TL;DR

This paper explores the structure of the Pick kernel for Schur functions on the polydisk, revealing it can be decomposed into two shift-invariant kernels, thus extending known positivity properties.

Contribution

It demonstrates that the Pick kernel on the polydisk can be decomposed into two shift-invariant kernels, revealing deeper structural properties.

Findings

Pick kernel always positive semi-definite on the polydisk

Kernel can be split into two shift-invariant kernels

Reveals new structural insights into Schur functions

Abstract

A certain kernel (sometimes called the Pick kernel) associated to Schur functions on the disk is always positive semi-definite. A generalization of this fact is well-known for Schur functions on the polydisk. In this article, we show that the Pick kernel on the polydisk has a great deal of structure beyond being positive semi-definite. It can always be split into two kernels possessing certain shift invariance properties.