The Bergman Shift Operator on Polynomial Lemniscates

TL;DR

This paper explores the connection between the Bergman shift operator and measures supported on polynomial lemniscates, providing new asymptotic results for such measures.

Contribution

It introduces novel results linking the Bergman shift operator to polynomial lemniscates and derives ratio asymptotics for measures supported on these curves.

Findings

Established a relationship between the Bergman shift operator and polynomial lemniscates

Derived ratio asymptotics for measures supported on polynomial lemniscates

Extended understanding of orthogonal polynomials on complex curves

Abstract

We investigate the relationship between the Bergman shift operator and the support of the corresponding measure. We pay special attention to the situation when the measure of orthogonality is concentrated on a polynomial lemniscate. As an application of our new results, we obtain a ratio asymptotic result for a wide variety of measures supported on polynomial lemniscates.