Scalar and matrix Riemann-Hilbert approach to the strong asymptotics of Padé approximants and complex orthogonal polynomials with varying weight
type: Item
instance of: scholarly article
title: Scalar and matrix Riemann-Hilbert approach to the strong asymptotics of Padé approximants and complex orthogonal polynomials with varying weight
zbMATH Open document ID: 1061.30035
author: Walter Van Assche
author: A. I. Aptekarev
published in: Journal of Approximation Theory
publication date: 2004-10-28T00:00:00Z
review text: An important development in the asymptotic theory of orthogonal polynomials is the introduction of methods based on matrix Riemann-Hilbert problems by \textit{P. Deift} et al. in [Comm. Pure Appl. Math. 52, 1335--1425, (1999; Zbl 0944.42013), 1491--1552 (1999; Zbl 1026.42024)] and \textit{P. Bleher} and \textit{A. Its} [Ann. Math. (2) 150, 185--266 (1999; Zbl 0956.42014)]. These papers were motivated by universality questions from random matrix theory. Boundary value problems on Riemann surfaces appeared in earlier works of \textit{J. Nuttall} [J. Approx. Theory 42, 299--386 (1984; Zbl 0565.41015)] and Stahl on Padé approximation. These boundary value problems can be considered as vector Riemann-Hilbert problems in contrast to the more recent matrix Riemann-Hilbert problems. In the present paper, the authors explain both methods and its relation to the strong asymptotics of Padé approximants to Markov functions. The boundary value problem on a Riemann surface is presented with some new ideas due to \textit{S. P. Suetin} [Mat. Sb. 191, No. 9, 81--114 (2000; Zbl 0980.41015)]. The matrix Riemann-Hilbert analysis is illustrated by a model problem of polynomials that are orthogonal on \([-1,1]\) with respect to a complex-valued weight function with square-root singularities at both end-points. These special weight functions have the advantage that the end-point analysis becomes trivial, which results in an easier and transparent exposition of the method.
Mathematics Subject Classification ID: 30E25
Mathematics Subject Classification ID: 30E10
Mathematics Subject Classification ID: 30E15
Mathematics Subject Classification ID: 41A21
Mathematics Subject Classification ID: 41A60
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 2111897
zbMATH Keywords: strong asymptotics
zbMATH Keywords: Padé approximants
zbMATH Keywords: Riemann-Hilbert problems
reviewed by: Arno B. J. Kuijlaars
MaRDI profile type: Publication
full work available at URL: j.jat.2004.06.001
OpenAlex ID: W1992143608
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DOI: 10.1016/J.JAT.2004.06.001
description: scientific article; zbMATH DE number 2111897