On some nonlinear spaces of approximating functions
type: Item
instance of: scholarly article
title: On some nonlinear spaces of approximating functions
zbMATH Open document ID: 0552.41012
DOI: 10.1016/0021-9045(84)90007-8
published in: Journal of Approximation Theory
publication date: 1984-01-01T00:00:00Z
review text: Let F be continuous and strictly positive on [0,\(\infty)\). Let \(V_ n\) denote the (non-linear) space of functions of the form F(\(\alpha\) x)P(x) where \(\alpha\geq 0\) and P is \(\pi_ n\). Various conditions on F are given to insure that \(V_ n\) is an existence space for C[a,b], \(0\leq a<b<\infty:\) i.e. so that every f in C[a,b] has a best approximation in \(V_ n\). In the simplest case the condition is that \(\lim_{x\to \infty}F(x)\) be finite and positive. In another result, the condition is that, for each \(x>1\), \(\lim_{\beta \to \infty}F(\beta x)/F(\beta)=\infty\). Another result involves positive rational functions. All proofs involve considering the best approximation in the space \(V_ n(\alpha)\), i.e. where \(\alpha\) is fixed, and then estimating the coefficients in the polynomial involved. Various examples are given to show that not much improvement in the hypotheses is possible. There are some remarks about the lack of uniqueness when \(F(x)=e^ x\), but these are not new.
Mathematics Subject Classification ID: 41A30
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 41A65
zbMATH DE Number: 3879461
zbMATH Keywords: Haar systems
zbMATH Keywords: examples
author: Richard P. Gosselin
MaRDI profile type: Publication
full work available at URL: 0021-9045(84)90007-8
OpenAlex ID: W2013318643
cites work: Chebyshev Approximation by Families with the Betweeness Property
cites work: Q5531436
cites work: Q5577758
cites work: Another property of Chebyshev polynomials
description: scientific article; zbMATH DE number 3879461