On the constrained Chebyshev approximation problem on ellipses
type: Item
instance of: scholarly article
title: On the constrained Chebyshev approximation problem on ellipses
zbMATH Open document ID: 0728.41023
DOI: 10.1016/0021-9045(90)90054-T
author: B. Fischer
author: Roland W. Freund
published in: Journal of Approximation Theory
publication date: 1990-01-01T00:00:00Z
review text: The following constrained Chebyshev approximation problem on ellipses is considered: \[ D_ n(r,c):=\min_{p\in \prod_ n,p(c)=1}\max_{z\in {\mathcal E}_ r}| p(z)|, \] where \(\prod_ n\) denotes the set of all complex polynomials of degree at most n \[ {\mathcal E}_ r=\{z\in {\mathbb{C}};| z-1| +| z+1| \leq r+\frac{1}{r}\}r\geq 1, \] is an ellipse in the complex plane \({\mathbb{C}}\) with foci at \(\pm 1\), and it is always assumed that \(c\in {\mathbb{C}}\setminus {\mathcal E}_ r\). The solution of (1) is classical for the case \(r=1\), of the line segment \({\mathcal E}_ 1=[-1,1]\) real c: \(P_ n(z,r,c)=T_ n(z)/T_ n(c),\) where \(T_ n\) is the nth Chebychev polynomial. In this paper the solution of (1) on \({\mathcal E}_ r\) is characterized, founded and studied. This problem were also studied by \textit{G. Opfer} and \textit{G. Schober} [Linear algebra appl. 58, 343-361 (1984; Zbl 0565.65012)] for \(n=1\) with \({\mathcal E}\subset {\mathbb{C}}\) any compact set not containing c.
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 41A29
Mathematics Subject Classification ID: 30E10
zbMATH DE Number: 4202989
zbMATH Keywords: Chebyshev approximation problem on ellipses
reviewed by: D. Zarnadze
MaRDI profile type: Publication
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full work available at URL: 0021-9045(90)90054-t
OpenAlex ID: W2063392327
description: scientific article; zbMATH DE number 4202989