The Hessenberg matrix and the Riemann mapping

TL;DR

This paper links the asymptotic behavior of Hessenberg matrices from orthogonal polynomials to the Riemann mapping function, providing a method to approximate the mapping from matrix entries.

Contribution

It establishes that the limit of the Hessenberg matrix is characterized by the Riemann mapping function, enabling approximation of the conformal map from matrix data.

Findings

Hessenberg matrix asymptotics relate to the Riemann mapping function.

The limit operator's symbol is the Riemann mapping function on the unit circle.

Method to approximate the Riemann mapping from Hessenberg matrix entries.

Abstract

We consider a Jordan arc \Gamma in the complex plane \mathbb{C} and a regular measure \mu whose support is \Gamma . We denote by D the upper Hessenberg matrix of the multiplication by z operator with respect to the orthonormal polynomial basis associated with \mu . We show in this work that, if the Hessenberg matrix D is uniformly asymptotically Toeplitz, then the symbol of the limit operator is the restriction to the unit circle of the Riemann mapping function \phi(z) which maps conformally the exterior of the unit disk onto the exterior of the support of the measure \mu . We use this result to show how to approximate the Riemann mapping function for the support of \mu from the entries of the Hessenberg matrix D.