A simple characterization of weighted Sobolev spaces with bounded multiplication operator
type: Item
instance of: scholarly article
title: A simple characterization of weighted Sobolev spaces with bounded multiplication operator
zbMATH Open document ID: 1221.41007
published in: Journal of Approximation Theory
publication date: 2008-07-31T00:00:00Z
review text: The definition of weighted Sobolev spaces introduced by \textit{A. Kufner} and \textit{B. Opic} [Commentat. Math. Univ. Carol. 25, 537--554 (1984; Zbl 0557.46025)] is extended to a wider class of vectorial measures, with the property of making them a Banach space. In this context, the author obtains a characterization for the boundedness of the multiplication operator \(\mathcal{M}f(x)=xf(x)\), a condition which in turn implies that the zeros of Sobolev orthogonal polynomials are bounded by \(\|\mathcal{M}\|\). Following the steps of Kufner--Opic, it is introduced a class \(B_ p(A)\) of weight functions in the set \(A\) whose \(1/(1-p)\)-th powers are locally integrable. The main result of the paper adds the assumption that the weights are comparable to a piecewise monotone function. Several examples are given at the end of the article.
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 46G10
zbMATH DE Number: 5306487
zbMATH Keywords: weighted Sobolev spaces
zbMATH Keywords: multiplication operator
zbMATH Keywords: location of zeros
zbMATH Keywords: Sobolev orthogonal polynomials
author: José M. Rodríguez
reviewed by: J. Illán
MaRDI profile type: Publication
full work available at URL: j.jat.2008.01.003
OpenAlex ID: W2023774095
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DOI: 10.1016/J.JAT.2008.01.003
description: scientific article; zbMATH DE number 5306487