Asymptotic behavior and zero distribution of Carleman orthogonal polynomials

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.