Poles and alternation points in real rational Chebyshev approximation
type: Item
instance of: scholarly article
title: Poles and alternation points in real rational Chebyshev approximation
zbMATH Open document ID: 1057.41008
DOI: 10.1007/BF03321033
author: R. Grothmann
author: R. K. Kovacheva
author: H.-P. Blatt
published in: Computational Methods and Function Theory
publication date: 2004-11-01T00:00:00Z
review text: The concept of alternation sets is well known in the theory of best polynomial approximation and the extension to real rational approximation has been studied extensively. Let \(f\in C[-1,1]\) be real valued and let \({\mathcal R}_{n,m}\) denote the set of rational functions of the form \(r=p/q\), where \(q\not\equiv 0\) and \(p\) and \(q\) are polynomials of degrees at most \(n\) and \(m\) respectively. For each pair \((n,m)\) of nonnegative integers there then exists a unique function \(r^{\ast}_{n,m}\in{\mathcal R}_{n,m}\) that satsfies \[ \| f-r^{\ast}_{n,m}\| <\| f-r\| \text{ for all }r\in{\mathcal R}_{n,m}, r\not= r^{\ast}_{n,m}, \] where \(\| \cdot \| \) is the sup-norm on \(C[-1,1]\). Write \(r^{\ast}_{n,m}=p^{\ast}_n/ q^{\ast}_m\), with numerator and denominator having no common factors, and define the \textit{defect} by \[ d_{n,m}=\min\{n-\deg p^{\ast}_n, m-\deg q^{\ast}_m\}. \] Then there exist \(\delta=m+n+2-d_{n,m}\) points \(x_k^{(n,m)}\) on the closed interval \([-1,1]\) \[ -1\leq x_1^{(n,m)}<\cdots <x_{\delta}^{(n,m)}\leq 1 \] satisfying \[ \lambda_{n,m}(-1)^k(f-r^{\ast}_{n,m}) (x_k^{(n,m)})=\| f-r^{\ast}_{n,m}\| ,\quad 1\leq k\leq \delta, \] where \(\lambda_{n,m}=\pm 1\), fixed; this is called an alternation set (in general not unique). For the sequel, let for each pair \((n,m)\) an arbitrary, fixed alternation set for the best approximation to \(f\) be given by \[ A_{n,m}=A_{n,m}(f):= \{x_k^{(n,m)}\}_{k=1}^{\delta}. \] In the paper of \textit{P. B. Borwein}, \textit{A. Króo}, \textit{R. Grothmann}, \textit{E. B. Saff}, Proc. Am. Math. Soc. 105, No.4, 881-888 (1989; Zbl 0688.41018)], it is shown that a subsequence of alternation sets is dense in \([-1,1]\) whenever \(n/m\rightarrow\kappa,\;\kappa>1\). In ``Behavior of alternation points in best rational approximation'', [Acta Appl. Math. 33, No.2-3, 195-210 (1993; Zbl 0801.41021)], the connection between denseness and the number of poles of \(r^{\ast}_{n,m}\) that lie outside an \(\varepsilon\)-neig…
reviewed by: Marcel G. de Bruin
Mathematics Subject Classification ID: 41A20
Mathematics Subject Classification ID: 41A50
zbMATH DE Number: 2112934
zbMATH Keywords: rational approximation
zbMATH Keywords: best approximation
zbMATH Keywords: alternation points
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf03321033
OpenAlex ID: W2049960820
description: scientific article; zbMATH DE number 2112934