A constrained Nevanlinna-Pick interpolation problem for matrix-valued functions

TL;DR

This paper extends the necessary and sufficient conditions for a constrained Nevanlinna-Pick interpolation problem to matrix-valued functions, broadening the classical theory to more complex domains and settings.

Contribution

It generalizes existing results to matrix-valued functions and explores analogies with interpolation on finitely-connected planar domains.

Findings

Established conditions for matrix-valued interpolation with derivative constraints

Extended classical results to more general domains and matrix settings

Connected the theory to Nevanlinna-Pick problems on finitely-connected domains

Abstract

Recent results of Davidson-Paulsen-Raghupathi-Singh give necessary and sufficient conditions for the existence of a solution to the Nevanlinna-Pick interpolation problem on the unit disk with the additional restriction that the interpolant should have the value of its derivative at the origin equal to zero. This concrete mild generalization of the classical problem is prototypical of a number of other generalized Nevanlinna-Pick interpolation problems which have appeared in the literature (for example, on a finitely-connected planar domain or on the polydisk). We extend the results of Davidson-Paulsen-Raghupathi-Singh to the setting where the interpolant is allowed to be matrix-valued and elaborate further on the analogy with the theory of Nevanlinna-Pick interpolation on a finitely-connected planar domain.