Mathematical Sciences Faculty Publications

Title

Asymptotic Behavior and Zero Distribution of Carleman Orthogonal Polynomials

Article

2010

Publication Source

Journal of Approximation Theory

162

11

1982-2003

yes

Abstract

Let $L$ be an analytic Jordan curve and let $p_n(z)$ 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 $lim_{n→∞} p_n(z)/ (\sqrt{(n + 1)/π} [φ(z)]^n) = φ′(z)$ locally uniformly on a certain open neighborhood of the closed exterior of $L$, where $φ$ 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 this fact 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.

Keywords

Orthogonal polynomials, Asymptotic behavior, Zeros of polynomials, Conformal maps

Disciplines

Analysis | Mathematics

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