Asymptotic behavior and zero distribution of Carleman orthogonal polynomials

JOURNAL OF APPROXIMATION THEORY

卷:162

Article

Dragnev, Peter¹  Mina-Diaz, Erwin² 

[1] Indiana Purdue Univ Ft Wayne, Dept Math Sci, Ft Wayne, IN 46805 USA

[2] Univ Mississippi, Dept Math, University, MS 38677 USA

关键词: Orthogonal polynomials;    Asymptotic behavior;    Zeros of polynomials;    Conformal maps;   

DOI  :  10.1016/j.jat.2010.05.006

来源: Elsevier

【 摘 要 】

Let L be an analytic Jordan curve and let {p(n)(z)}(n=0)(infinity) 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 ->infinity) pn(z)/root(n + 1)/pi vertical bar phi(z)vertical bar(n) = phi'(z) locally uniformly on a certain open neighborhood of the closed exterior of L, where 0 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 (1) 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. (C) 2010 Elsevier Inc. All rights reserved.

【 授权许可】

Free