Ratio Asymptotics, Hessenberg Matrices, and Weak Asymptotic Measures

TL;DR

This paper explores the link between ratio asymptotics of orthogonal polynomials, the asymptotic behavior of the Bergman shift operator, and weak asymptotics of measures on complex supports, revealing structural relationships.

Contribution

It establishes a clear connection between ratio asymptotics and the asymptotic Toeplitz nature of the Bergman shift operator for measures supported on complex sets.

Findings

Orthogonal polynomials with ratio asymptotics correspond to asymptotically Toeplitz Bergman shift operators.

The work links weak asymptotics of measures to operator asymptotics.

Provides a framework for understanding polynomial and measure asymptotics in complex analysis.

Abstract

We discuss the relationship between ratio asymptotics for general orthogonal polynomials and the asymptotics of the associated Bergman shift operator. More specifically, we consider the case in which a measure is supported on an infinite compact subset of the complex plane. We show that there is a straightforward connection between the corresponding orthonormal polynomials exhibiting ratio asymptotics and the corresponding Bergman shift operator being asymptotically Toeplitz. We also discuss a connection to the weak asymptotics of the measures derived from the orthonormal polynomials.