Asymptotic behavior of orthogonal polynomials corresponding to a measure with infinite discrete part off a curve
type: Item
instance of: scholarly article
title: Asymptotic behavior of orthogonal polynomials corresponding to a measure with infinite discrete part off a curve
zbMATH Open document ID: 0882.42014
author: Rachid Benzine
published in: Journal of Approximation Theory
publication date: 1998-02-16T00:00:00Z
review text: In this paper, the author obtains Szegö type asymptotics for orthogonal polynomials associated with a positive Borel measure \(\alpha\) on a rectifiable Jordan curve \(E\) in the complex plane. The measure has the form \(\alpha= \beta+\gamma\), where \(\beta\) satisfies the relevant form of Szegö's condition on the curve, and \(\gamma\) is a pure jump measure. The mass points of \(\gamma\) lie in a countable sequence of points \(\{z_k\}^\infty_{k=1}\) outside \(E\); technical conditions are placed involving these points, the conformal map \(\Phi\) of the exterior of \(E\) onto the exterior of the unit ball, and the monic orthonormal polynomial for \(d\alpha\), \(P_n(d\alpha, z)=z^n+\cdots\). The author proves the asymptotic formula \[ P_n(d\alpha,z)= (\text{cap}(E)\Phi(z))^n[\varphi^*(z) B(z)+o(1)] \] uniformly on compact subsets of the exterior of \(E\). Here \(\text{cap}(E)\) is the logarithmic capacity of \(E\) and \(B(z)\) is a Blaschke product involving the mass points \(\{z_k\}^\infty_{k=1}\) and the conformal map \(\Phi\). Moreover, \(\varphi^*(z)\) is the solution of an extremal problem for the Hardy 2 space associated with the measure \(\beta\) and the exterior of \(E\).
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 41A60
zbMATH DE Number: 1019194
zbMATH Keywords: Szegö type asymptotics for orthogonal polynomials
zbMATH Keywords: Szegö condition
zbMATH Keywords: jump measure
reviewed by: Doron S. Lubinsky
MaRDI profile type: Publication
full work available at URL: jath.1997.3041
OpenAlex ID: W2117828454
cites work: Q5815837
cites work: Q4732888
cites work: Q3874458
cites work: Q5835456
cites work: Q5549176
cites work: Q4061445
DOI: 10.1006/JATH.1997.3041
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description: scientific article; zbMATH DE number 1019194