Pick interpolation on the polydisc: small families of sufficient kernels

TL;DR

This paper characterizes all data for Pick interpolation on the unit polydisc in several complex variables using a family of kernels, advancing understanding of boundary behavior and kernel structures.

Contribution

It provides a comprehensive characterization of Pick interpolation data on the polydisc via parametrized positive-definite kernels, employing duality and approximation techniques.

Findings

Characterization of data admitting a $ ext{D}$-valued interpolant

Identification of kernel families parametrized by polynomials

Analysis of boundary points where solutions are not unimodular

Abstract

We give a solution to Pick's interpolation problem on the unit polydisc in $\mathbb{C}^n$, $n\geq 2$, by characterizing all interpolation data that admit a $\mathbb{D}$-valued interpolant, in terms of a family of positive-definite kernels parametrized by a class of polynomials. This uses a duality approach that has been associated with Pick interpolation, together with some approximation theory. Furthermore, we use duality methods to understand the set of points on the $n$-torus at which the boundary values of a given solution to an extremal interpolation problem are not unimodular.