The convergence of Padé approximants to functions with branch points
type: Item
instance of: scholarly article
title: The convergence of Padé approximants to functions with branch points
zbMATH Open document ID: 0896.41009
published in: Journal of Approximation Theory
publication date: 1998-01-06T00:00:00Z
review text: The dates of receipt of the original and revised versions of this paper tell a story: the original was received in 1983, and the revised in 1997. In the fourteen years in between, the author's ideas have revolutionized not only the convergence theory of Padé approximation, but also the study of complex orthogonal polynomials, rational approximation, and numerous extremal problems. The reasons for the delay are administrative; it is a delight to finally have this polished paper in print. He presents here one approach to analyzing the convergence of sequences of Padé approximants to functions with branchpoints. There are two classes of functions considered. The first is analytic but not necessarily single-valued in the plane except for singularities in a set of capacity \(0\). The second class of functions is analytic in a domain possessing a symmetry property. In both cases, the author establishes convergence in capacity, with the exact rate, of sequences of diagonal Padé approximants in a certain domain. Outside this domain, there is divergence in capacity. The actual domain of convergence is completely characterized, and in the case of functions whose branchpoints have a simple structure, may be described with the aid of quadratic differentials. The proofs rely heavily on asymptotic properties of orthogonal polynomials in the complex plane, with respect to an ``inner product'' that does not have the usual positivity properties, and employ sophisticated potential theory. It is good to have this classic in print: for many years it and its ideas were quoted and the question always was: but where did it or will it appear? Now, we may all quote actual page numbers.
Mathematics Subject Classification ID: 41A21
Mathematics Subject Classification ID: 30E10
zbMATH DE Number: 1100511
zbMATH Keywords: potential theory
zbMATH Keywords: Padé approximation
zbMATH Keywords: branchpoints
zbMATH Keywords: quadratic differentials
author: Herbert Stahl
reviewed by: Doron S. Lubinsky
MaRDI profile type: Publication
full work available at URL: jath.1997.3141
OpenAlex ID: W2052680602
cites work: Q3739430
cites work: A LOCAL CONDITION OF SINGLE-VALUEDNESS OF ANALYTIC FUNCTIONS
cites work: Q3788439
cites work: Q3255547
cites work: Rational interpolation and best rational approximation
cites work: Q5666837
cites work: Divergence of complex rational approximations
cites work: Toeplitz matrix techniques and convergence of complex weight Padé approximants
cites work: On convergence in capacity
cites work: Q5818666
cites work: Q3872126
cites work: Asymptotics of diagonal Hermite-Padé polynomials
cites work: Orthogonal polynomials and Padé approximants associated with a system of arcs
cites work: Padé approximants and convergence in capacity
cites work: ON THE CONVERGENCE OF PADÉ APPROXIMANTS IN CLASSES OF HOLOMORPHlC FUNCTIONS
cites work: Q3701896
cites work: Extremal domains associated with an analytic function I
cites work: Extremal domains associated with an analytic function II
cites work: The structure of extremal domains associated with an analytic function
cites work: Divergence of diagonal Padé approximants and the asymptotic behavior of orthogonal polynomials associated with nonpositive measures
cites work: Orthogonal polynomials with complex-valued weight function. I
cites work: Orthogonal polynomials with complex-valued weight function. II
cites work: On the convergence of generalized Padé approximants
cites work: Q5289656
DOI: 10.1006/JATH.1997.3141
description: scientific article; zbMATH DE number 1100511