Multipoint Padé-type approximants. Exact rate of convergence
type: Item
instance of: scholarly article
title: Multipoint Padé-type approximants. Exact rate of convergence
zbMATH Open document ID: 0896.41010
published in: Constructive Approximation
publication date: 1998-04-20T00:00:00Z
full work available at URL: http://hdl.handle.net/10016/6368
review text: In this interesting paper, the authors consider sequences of interpolating rational functions to Markov-type analytic functions, that is functions of the form \[ f(z):= \int^1_{-1} {d\mu(x)\over z-x},\quad z\in\mathbb{C}\backslash[- 1,1], \] where \(\mu\) is a finite positive Borel measure on \([-1,1]\). The interpolation points \(\{\alpha_{n,j}\}\) are assumed to be symmetric with respect to the real line, and some of the poles of the interpolating rational functions may be taken as zeros of a monic polynomial \(B_n\) of degree \(m(n)\leq n\), which is of one sign in \((-1,1)\). The rest of the poles of the interpolating rationals are determined by interpolation conditions. Under the assumption that \(\mu'>0\) a.e. in \((-1,1)\) and under suitable asymptotic assumptions on the distribution of the \(\{\alpha_{n,j}\}\) and the zeros of \(B_n\), the authors obtain the exact asymptotic rate of convergence of the interpolants. Then applications are given to the convergence rates of approximate quadratures. The proofs involve sophisticated potential theory, and the results strengthen and sharpen earlier work of these authors as well as of other authors including A. Ambroladze, L. Karlberg, H. Wallin. This is essential reading for researchers in the convergence theory of rational interpolation.
Mathematics Subject Classification ID: 41A21
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 30E10
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 65D30
zbMATH DE Number: 1143073
zbMATH Keywords: Markov functions
zbMATH Keywords: approximate quadratures
zbMATH Keywords: rational interpolation
author: Francisco Cala Rodríguez
author: Guillermo López Lagomasino
reviewed by: Doron S. Lubinsky
MaRDI profile type: Publication
OpenAlex ID: W1982022705
DOI: 10.1007/S003659900074
description: scientific article; zbMATH DE number 1143073