Asymptotics for Hessenberg matrices for the Bergman shift operator on Jordan regions

TL;DR

This paper investigates the asymptotic behavior of Hessenberg matrices associated with the Bergman shift operator on Jordan domains, revealing their relation to Toeplitz matrices and enabling shape recovery from moments.

Contribution

It establishes a connection between Hessenberg matrices for Bergman shift operators and Toeplitz matrices via conformal maps, based on strong asymptotics of Bergman polynomials.

Findings

Describes the asymptotic relation between Hessenberg and Toeplitz matrices.

Provides an algorithm for shape reconstruction of Jordan domains from area moments.

Analyzes the convergence properties of the associated matrices.

Abstract

Let G be a bounded Jordan domain in the complex plane and consider the infinite upper Hessenberg matrix M associated with the Bergman orthogonal polynomials of G. This matrix represents the Bergman shift operator of G. The main purpose of the paper is to describe and analyze a close relation between M and the Toeplitz matrix with symbol the normalized conformal map of the exterior of the unit circle onto the complement of the closure of G. Our results are based on the strong asymptotics of the Bergman polynomials. As an application, we describe and analyze an algorithm for recovering the shape of G from its area moments.