Weighted Weierstrass' theorem with first derivatives
type: Item
instance of: scholarly article
title: Weighted Weierstrass' theorem with first derivatives
zbMATH Open document ID: 1221.46028
author: Ana Portilla
author: José M. Rodríguez
author: Eva Tourís
author: Yamilet Quintana
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-07-19T00:00:00Z
full work available at URL: http://hdl.handle.net/10016/6443
review text: Let \(w:\mathbb R\to[0,+\infty]\) a measurable function and \(L^\infty(w)\) the space of measurable functions \(f:\mathbb R\to\mathbb R\) such that \[ \|f\|:\text{ess\,sup}|f(x)|w(t)\mid< +\infty,\;x\in\mathbb R, \] with the convention \(0 \cdot\infty=0\). In this paper, the authors characterize the closure of \(C(\mathbb R)\) in \(L^\infty(w)\) for any weight \(w\) and as a consequence they obtain the set of functions which can be approximated in \(L^\infty(w)\) by polynomials for any weight with compact support. Moreover, under suitable conditions, they give analogous characterizations for the set of functions which can be approximated by smooth functions in the Sobolev space \(W^{1,\infty}(w_0,w_1)\).
reviewed by: Delfina Roux
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 5173014
zbMATH Keywords: weighted spaces \(L^\infty\) and \(w^{1,\infty}\), approximation by polynomials
zbMATH Keywords: approximation by smooth functions
MaRDI profile type: Publication
OpenAlex ID: W2075391679
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DOI: 10.1016/J.JMAA.2006.12.066
description: scientific article; zbMATH DE number 5173014