Asymptotic behaviour of orthogonal polynomials on the unit circle with asymptotically periodic reflection coefficients. II: Weak asymptotics
type: Item
instance of: scholarly article
title: Asymptotic behaviour of orthogonal polynomials on the unit circle with asymptotically periodic reflection coefficients. II: Weak asymptotics
zbMATH Open document ID: 0971.42013
author: Franz Peherstorfer
author: R. Steinbauer
published in: Journal of Approximation Theory
publication date: 2001-10-23T00:00:00Z
review text: Orthogonal polynomials on the unit circle are completely determined by their reflection coefficients \(a_n = P_{n+1}(0)\) which appear as recurrence coefficients in the recurrence relation for the monic polynomials \(P_n(z)\). In this paper the authors assume that the reflection coefficients are asymptotically periodic with period \(N\), i.e., \(\lim_{k \to \infty} a_{kN+j} = a_j^0\) for \(j=0,1,\ldots,N-1\). First, the orthogonal polynomials with periodic reflection coefficients are analyzed in Section 2, where known results of Geronimus are recalled. The support of the orthogonality measure consists of at most \(N\) arcs on the unit circle and at most a finite number of mass points outside these arcs. Then, in Section 3, the orthogonal polynomials with asymptotically periodic reflection coefficients are investigated, using the periodic case as a comparison system. The results in this paper deal with weak asymptotics, i.e., the asymptotics of integrals of the form \(\int g(\theta) |\Phi_{kN+j}(e^{i\theta})|^2 d\sigma\) when \(k \to \infty\) and \(\Phi_n\) the orthonormal polynomials (Theorem 3). The authors also deal with weak asymptotics of the Christoffel function (Theorem 4). [For Part I see J. Approximation Theory 88, No. 3, 316-353 (1997; Zbl 0889.42021)].
reviewed by: Walter Van Assche
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 30E15
Mathematics Subject Classification ID: 30C10
zbMATH DE Number: 1495765
zbMATH Keywords: orthogonal polynomials on the unit circle
zbMATH Keywords: reflection coefficients
zbMATH Keywords: asymptotics
zbMATH Keywords: periodic coefficients
MaRDI profile type: Publication
full work available at URL: jath.2000.3450
OpenAlex ID: W2943482684
cites work: Ratio asymptotics for polynomials orthogonal on arcs on the unit circle
cites work: Q5618011
cites work: Q5835993
cites work: Q5835453
cites work: Q5835454
cites work: Q5523124
cites work: Perturbation of orthogonal polynomials on an arc of the unit circle
cites work: Simplified Treatment of Integrals of Cauchy Type, the Hilbert Problem and Singular Integral Equations. Appendix: Poincaré-Bertrand Formula
cites work: Extensions of Szegö's theory of orthogonal polynomials. II
cites work: Q5736329
cites work: Gauss-Chebyshev quadrature formulas
cites work: A special class of polynomials orthogonal on the unit circle including the associated polynomials
cites work: Orthogonal polynomials on arcs of the unit circle. I
cites work: Orthogonal polynomials on arcs of the unit circle. II: Orthogonal polynomials with periodic reflection coefficients
cites work: Asymptotic behaviour of orthogonal polynomials on the unit circle with asymptotically periodic reflection coefficients
cites work: Strong Asymptotics of Orthonormal Polynomials with the Aid of Green's Function
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: On orthogonal polynomials
cites work: Christoffel functions and Turán determinants on several intervals
cites work: Extremal polynomials associated with a system of curves in the complex plane
DOI: 10.1006/JATH.2000.3450
Recommended article: Strong Asymptotics of Orthonormal Polynomials with the Aid of Green's Function
Recommended article: Asymptotic behaviour of orthogonal polynomials on the unit circle with asymptotically periodic reflection coefficients
Recommended article: Orthogonal polynomials on the circumference and arcs of the circumference
Recommended article: Perturbation of orthogonal polynomials on an arc of the unit circle. II
Recommended article: Q4941037
Recommended article: Asymptotics of polynomials orthogonal with respect to varying measures
Recommended article: Q3398763
Recommended article: Fine structure of the zeros of orthogonal polynomials III: Periodic recursion coefficients
Recommended article: Christoffel functions and Turán determinants on several intervals
Recommended article: A survey of general orthogonal polynomials for weights on finite and infinite intervals
description: scientific article; zbMATH DE number 1495765