Asymptotics of polynomials orthogonal with respect to varying measures
type: Item
instance of: scholarly article
title: Asymptotics of polynomials orthogonal with respect to varying measures
zbMATH Open document ID: 0669.42012
DOI: 10.1007/BF01889607
published in: Constructive Approximation
publication date: 1989-01-01T00:00:00Z
review text: This very important paper deals with orthogonal polynomials \(\phi_{n,m}(z)\) \((m=0,1,2,...)\) on the unit circle with respect to a measure \(d\sigma_ n(\theta)=d\sigma (\theta)/| W_ n(e^{i\theta})|^ 2\), where \(W_ n\) is a polynomial of degree n whose zeros lie on the closed unit disk \(\{\) \(| z| \leq 1\}\) and \(\sigma '>0\) almost everywhere on the unit circle. The asymptotic behaviour of the ratio \(\phi_{n,n+k+1}(z)/\phi_{n,n+k}(z)\) for \(n\to \infty\) is examined for \(| z| \geq 1\), the behaviour of the reflection coefficients and various strong and weak asymptotic results are given. The results are extended to complex measures on the unit circle. The transition to the real line is made and related asymptotic behaviour of orthogonal polynomials on [-1,1] with varying measures is given. The special case \(W_ n(z)=z^ n\) gives the famous results of E. A. Rakhmanov for orthogonal polynomials on the unit circle (and on the real line) with \(\sigma '>0\) almost everywhere. Power asymptotic behaviour of orthogonal polynomials with varying measures is given by \textit{G. López} [Lect. Notes Math. 1329, 255-260 (1988; Zbl 0652.42014)].
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 41A25
zbMATH DE Number: 4096123
zbMATH Keywords: Power asymptotic behaviour
zbMATH Keywords: orthogonal polynomials with varying measures
author: Guillermo López Lagomasino
reviewed by: Walter Van Assche
MaRDI profile type: Publication
cites work: Q5618011
cites work: A class of orthogonal polynomials
cites work: ON CONVERGENCE OF PADÉ APPROXIMANTS FOR SOME CLASSES OF MEROMORPHIC FUNCTIONS
cites work: ON MARKOV'S THEOREM FOR MULTIPOINT PADÉ APPROXIMANTS
cites work: Asymptotics for the ratio of leading coefficients of orthonormal polynomials on the unit circle
cites work: Strong and Weak Convergence of Orthogonal Polynomials
cites work: ON THE ASYMPTOTICS OF THE RATIO OF ORTHOGONAL POLYNOMIALS
cites work: ON THE ASYMPTOTICS OF THE RATIO OF ORTHOGONAL POLYNOMIALS. II
cites work: Orthogonal polynomials with complex-valued weight function. I
cites work: Orthogonal polynomials with complex-valued weight function. II
Recommended article: Q3145717
Recommended article: Some asymptotic properties for orthogonal polynomials with respect to varying measures
Recommended article: Relative asymptotic behavior of sequences of orthogonal polynomials on the real line.
Recommended article: Q5289656
Recommended article: Q3831445
Recommended article: Extensions of Szegö's theory of orthogonal polynomials. II
Recommended article: Perturbation of orthogonal polynomials on an arc of the unit circle. II
Recommended article: Asymptotics for the ratio of leading coefficients of orthonormal polynomials on the unit circle
Recommended article: Q3838538
Recommended article: Strong Asymptotics of the Orthogonal Polynomials with Respect to a Measure Supported on the Plane
description: scientific article; zbMATH DE number 4096123