Complete Nevanlinna-Pick kernels

TL;DR

This paper provides a new characterization and representation of complete Nevanlinna-Pick kernels, showing their equivalence with matrix-valued properties and their relation to a universal kernel.

Contribution

It introduces a novel treatment of the characterization of complete Nevanlinna-Pick kernels and demonstrates their universal restriction property.

Findings

Equivalence of matrix-valued and vector-valued Nevanlinna-Pick properties

Representation of all complete Nevanlinna-Pick kernels

All such kernels are restrictions of a universal kernel

Abstract

We give a new treatment of Quiggin's and McCullough's characterization of complete Nevanlinna-Pick kernels. We show that a kernel has the matrix-valued Nevanlinna-Pick property if and only if it has the vector-valued Nevanlinna-Pick property. We give a representation of all complete Nevanlinna-Pick kernels, and show that they are all restrictions of a universal complete Nevanlinna-Pick kernel.