A note on the asymptotics of orthogonal polynomials on a complex arc: The case of a measure with a discrete part
type: Item
instance of: scholarly article
title: A note on the asymptotics of orthogonal polynomials on a complex arc: The case of a measure with a discrete part
zbMATH Open document ID: 0831.42015
published in: Journal of Approximation Theory
publication date: 1996-02-04T00:00:00Z
review text: Orthogonal polynomials with respect to a measure \(\mu= \mu_1+ \mu_2\) in the complex plane are studied, where \(\mu_1\) has support on a complex rectifiable are \(E\) and is absolutely continuous with respect to arc length, and \(\mu_2\) has finite support on \(N\) points in the exterior of \(E\). For \(E= [- 1,1]\) the asymptotic behaviour of the orthogonal polynomials is fairly well-known, but for a complex arc there is no recurrence relation and there are no nice properties of the zeros, so that the analysis is much harder. Also for a closed curve there are nice results, but the case of an arc is quite different. The author proposes to study the orthogonal polynomials using the extremal property, an approach previously developed by H. Widom in 1969. The main result is that for a sufficiently smooth arc and for a measure \(\mu_1\) satisfying a Szegö condition on \(E\), one can describe the strong asymptotic behaviour of the \(L_2(\mu)\)-norm of the monic orthogonal polynomials in terms of the capacity of \(E\), the asymptotic behaviour of the orthogonal polynomials for the measure \(\mu\) in \(L_2(\mu)\) and the strong asymptotics in the exterior of \(E\) with a limit function that vanishes at the \(N\) points in the support of \(\mu_2\).
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 30E10
zbMATH DE Number: 720671
zbMATH Keywords: extremal polynomials
zbMATH Keywords: Szegö theory
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: strong asymptotics
author: V. A. Kaliaguine
reviewed by: Walter Van Assche
MaRDI profile type: Publication
full work available at URL: jath.1995.1009
OpenAlex ID: W2485186781
DOI: 10.1006/JATH.1995.1009
description: scientific article; zbMATH DE number 720671