Relative Asymptotics of Orthogonal Polynomials for Perturbed Measures

TL;DR

This paper introduces polynomially small perturbations of measures in the context of orthogonal polynomials and establishes their relative asymptotic behavior and zero distribution, especially for Bergman polynomials.

Contribution

It defines PS perturbations of measures and derives new asymptotic results for orthogonal polynomials and their zeros under these perturbations.

Findings

Relative asymptotics for orthogonal polynomials under PS perturbations

Behavior of zeros of perturbed Bergman polynomials

New insights into measure perturbations in orthogonal polynomial theory

Abstract

In the context of orthogonal polynomials in the plane we introduce the notion of a polynomially small (PS) perturbation of a measure. In such a case we establish relative asymptotic results for the two sequences of the associated orthonormal polynomials. We also provide results dealing with the behaviour of the zeros of PS perturbations of area orthogonal (Bergman) polynomials.