Title

Asymptotic Behavior and Zero Distribution of Carleman Orthogonal Polynomials

Document Type

Article

Publication Date

2010

Publication Source

Journal of Approximation Theory

Volume

162

Issue

11

Inclusive pages

1982-2003

Peer Reviewed

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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