The three-point Pick-Nevanlinna interpolation problem on the polydisc

TL;DR

This paper characterizes the conditions under which a holomorphic interpolant exists on the unit polydisc for three-point data, focusing on rational inner functions and their factorization, advancing the understanding of multivariable interpolation.

Contribution

It provides a new characterization for three-point Pick-Nevanlinna interpolation on the polydisc, linking the problem to rational inner functions and their factorization.

Findings

Characterization of interpolant existence on the polydisc.

Reduction to finding a rational inner function satisfying positivity.

Results on factorization of rational inner functions.

Abstract

We give a characterization for the existence of a holomorphic interpolant on the unit polydisc $\mathbb{D}^n,$ $n\geq 2,$ for prescribed three-point Pick--Nevanlinna data. One of the key steps is a characterization for the existence of an interpolant that is a rational inner function on $\mathbb{D}^n.$ The latter reduces the search for a three-point interpolant to finding a single rational inner function that satisfies a type of positivity condition and arises from a polynomial of a very special form. This in turn relies on a pair of results, which are of independent interest, on the factorization of rational inner functions.