Asymptotic behavior and zero distribution of Carleman orthogonal polynomials

TL;DR

This paper extends Carleman's asymptotic results for orthogonal polynomials on Jordan curves, analyzing their zero distribution and providing concrete examples and numerical illustrations.

Contribution

It broadens the known asymptotic behavior of Carleman orthogonal polynomials to a maximal set including zero accumulation points.

Findings

Extended asymptotic validity to a maximal open set

Analyzed zero distribution of the polynomials

Provided numerical examples and illustrations

Abstract

Let $L$ be an analytic Jordan curve and let $\{p_n(z)\}_{n=0}^\infty$ be the sequence of polynomials that are orthonormal with respect to the area measure over the interior of $L$. A well-known result of Carleman states that \label{eq12} \lim_{n\to\infty}\frac{p_n(z)}{\sqrt{(n+1)/\pi} [\phi(z)]^{n}}= \phi'(z) locally uniformly on certain open neighborhood of the closed exterior of $L$, where $\phi$ is the canonical conformal map of the exterior of $L$ onto the exterior of the unit circle. In this paper we extend the validity of (\ref{eq12}) to a maximal open set, every boundary point of which is an accumulation point of the zeros of the $p_n$'s. Some consequences on the limiting distribution of the zeros are discussed, and the results are illustrated with two concrete examples and numerical computations.