Polynomial approximation using projections whose kernels contain the Chebyshev polynomials
type: Item
instance of: scholarly article
title: Polynomial approximation using projections whose kernels contain the Chebyshev polynomials
zbMATH Open document ID: 0659.41011
DOI: 10.1016/0021-9045(88)90026-3
author: B. L. Chalmers
author: George M. Phillips
author: Peter J. Taylor
published in: Journal of Approximation Theory
publication date: 1988-01-01T00:00:00Z
review text: The authors study projections of \(C^{(n+1)}[-1,1]\) onto the space \(P_ n\) of polynomials of degree \(\leq n\) with the property that the error in the Chebyshev norm can be estimated by \[ (*)\quad \| f-Pf\| \leq \frac{1}{2^ n(n+1)!}\| f^{(n+1)}\|. \] They find two necessary conditions one of which states that \(P(T_{n+1})=0\) for the Chebyshev polynomial \(T_{n+1}\) of degree \(n+1\). These conditions are not sufficient for (*). A complete characterization of projections satisfying (*) is given only in the case \(n=0\). It is well known that the projector \(P^*\) which assigns to f the polynomial \(P^*f\) interpolating f at the zeros of \(T_{n+1}\) satisfies (*). The authors show that all projections in a certain neighbourhood of \(P^*\) also satisfy (*). Reviewer's remarks: 1) In the proof of theorem 2.3 f should be replaced by f-p at several places. 2) In theorem 3.2 the condition on the measure must be strengthened by ``absolutely continuous measure \(\nu\) with density \(0\leq c(t)\leq 1''\). 3) Theorem 3.4 is wrong. Example 3.3 in the paper is a counterexample. The authors informed the reviewer that the conjecture on page 329 must be modified. A simple counterexample to the conjecture as stated in the paper would be for \(n=0\) the projector \(Pf=f(-1)+f(0)+f(1)-f(-1/2)-f(1/2).\)
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 41A17
zbMATH DE Number: 4077931
zbMATH Keywords: projections
zbMATH Keywords: Chebyshev norm
zbMATH Keywords: Chebyshev polynomial
reviewed by: Rudolf Wegmann
MaRDI profile type: Publication
full work available at URL: 0021-9045(88)90026-3
OpenAlex ID: W2031151577
cites work: Error estimates for least squares approximation by polynomials
cites work: Taylor-Like Remainder Formulas for Interpolation by Arbitrary Linear Functionals
cites work: Error of truncated Chebyshev series and other near minimax polynomial approximations
cites work: Q4070193
cites work: Polynomial approximation using equioscillation on the extreme points of Chebyshev polynomials
description: scientific article; zbMATH DE number 4077931