Ratio and relative asymptotics of polynomials orthogonal on an arc of the unit circle
type: Item
instance of: scholarly article
title: Ratio and relative asymptotics of polynomials orthogonal on an arc of the unit circle
zbMATH Open document ID: 0897.42016
published in: Journal of Approximation Theory
publication date: 1998-10-05T00:00:00Z
full work available at URL: http://hdl.handle.net/10016/6366
review text: In this interesting and important paper, the authors obtain ratio asymptotics for polynomials orthogonal with respect to a measure supported on an arc \(\gamma\) of the unit circle. Thus let \(\sigma\) be a finite positive Borel measure supported on \(\gamma\), and let \[ \varphi_n(\zeta)= \alpha_n\zeta^n+\cdots,\;\alpha_n>0, \] denote the \(n\)th orthonormal polynomial for \(\sigma\), so that \[ \int_{| z|= 1}\varphi_n(\zeta)\overline{\varphi_m(\zeta)} d\sigma(\zeta)= \delta_{mn},\;m,n\geq 0. \] The authors prove that \[ \lim_{n\to\infty} {\alpha_{n+ 1}\over\alpha_n}= {1\over\text{cap}(\gamma)}, \] where \(\text{cap}(\gamma)\) denotes the logarithmic capacity of \(\gamma\), and that \[ \lim_{n\to\infty} {\varphi_{n+ 1}(\zeta)\over \varphi_n(\zeta)}= G(\zeta) \] uniformly on compact subsets of \(\mathbb{C}\setminus\gamma\), where \(G\) denotes the conformal map of \(\mathbb{C}\setminus\gamma\) onto the complement of the unit ball. This extends a classic result of Rahmanov for the case where \(\gamma\) was the unit circle. Several applications and related results are presented. The topic treated by the authors is currently being actively investigated. This paper is undoubtedly one of the foundation papers for all subsequent research.
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 30E10
Mathematics Subject Classification ID: 30E15
zbMATH DE Number: 1146509
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: ratio asymptotics
zbMATH Keywords: logarithmic capacity
author: M. Bello Hernández
author: Guillermo López Lagomasino
reviewed by: Doron S. Lubinsky
MaRDI profile type: Publication
OpenAlex ID: W1977022777
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DOI: 10.1006/JATH.1997.3126
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description: scientific article; zbMATH DE number 1146509