Average widths of Sobolev classes on \(\mathbb{R}^ n\)
type: Item
instance of: scholarly article
title: Average widths of Sobolev classes on \(\mathbb{R}^ n\)
zbMATH Open document ID: 0802.41023
published in: Journal of Approximation Theory
publication date: 1994-02-17T00:00:00Z
review text: For \(p= (p_ 1,\dots, p_ n)\), \(1\leq p_ i\leq \infty\), \(i=1,\dots, n\), denote by \(L^ p (\mathbb{R}^ n)\) the Banach space of Lebesgue measurable functions on \(\mathbb{R}^ n\), with a mixed norm in the sense of \textit{A. Benedèk} and \textit{R. Panzone} [Duke Math. J. 28, 301-314 (1961; Zbl 0107.089)]. For a positive nondecreasing function \(\varphi\) on \((0,\infty)\), such that \(\varphi(a)\to \infty\) as \(a\to \infty\), the author defines the \(\varphi\)-average dimension of a subspace of \(L_ p (\mathbb{R}^ n)\) and, for \(\nu\geq 0\), the corresponding Kolmogorov \(\varphi\)-average \(\nu\)-width of a centrally symmetric subset of \(L_ p (\mathbb{R}^ n)\). In the case of the Sobolev class \(W_ p^ r (\mathbb{R}^ n)\) this width is finite if and only if \(\lim_{a\to\infty} (a^ n/ \varphi(a))<\infty\). Its asymptotic behaviour for \(a\to\infty\) is determined. In the case \(\varphi(a)= (2a)^ n\) the exact value of the average \(\nu\)-width of the unit ball of the subspace \(H_ 2^ \alpha (\mathbb{R}^ n)\), \(\alpha>0\), of \(L_ 2(\mathbb{R}^ n)\) is calculated.
Mathematics Subject Classification ID: 41A46
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 502998
zbMATH Keywords: Kolmogorov width
zbMATH Keywords: \(L^ p\)-spaces
zbMATH Keywords: Sobolev spaces
zbMATH Keywords: \(\nu\)-width
author: G. G. Magaril-Il'yaev
reviewed by: Stefan Cobzaş
MaRDI profile type: Publication
full work available at URL: jath.1994.1005
OpenAlex ID: W1993243023
DOI: 10.1006/JATH.1994.1005
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description: scientific article; zbMATH DE number 502998