The Multivariate Herglotz-Nevanlinna Class: Superresolution

TL;DR

This paper investigates the stability and uniqueness of solutions in multivariate holomorphic interpolation problems, focusing on superresolution and employing optimization and volume estimate techniques.

Contribution

It introduces a new analysis of solution continuity near unique solutions in multivariate interpolation, extending understanding of rational inner functions and automorphisms in complex domains.

Findings

Solutions are continuous near unique interpolation data points.

Superresolution results are established using optimization and volume estimates.

The approach connects interpolation problems with multivariable moment problems.

Abstract

Bounded holomorphic interpolation problems associated to finitely many data have, in general, distinct solutions. Uniqueness arises only in some convex extreme configurations. Rational inner functions in a polydisk are the best understood examples in this sense. We analyze the continuity of global solutions as functions of the finite interpolation data in neighbourhoods of elements distinguished by this uniqueness property. Our study covers rational inner or Cayley rational inner functions in the polydisk and automorphisms of the Euclidean ball. The proof of the main superresolution result is derived from optimization theory techniques and volume estimates of sublevel sets of real polynomials, both emerging from Markov's multivariable moment problem.