## 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

COinS

#### Link to Original Published Item

http://portal.acm.org/citation.cfm?id=1872746&CFID=5065127&CFTOKEN=85428852