Linear widths of a multivariate function space equipped with a Gaussian measure
type: Item
instance of: scholarly article
title: Linear widths of a multivariate function space equipped with a Gaussian measure
zbMATH Open document ID: 1073.41023
author: Chen Guanggui
author: Fang Gensun
published in: Journal of Approximation Theory
publication date: 2005-02-22T00:00:00Z
review text: This paper continues the investigation carried out in [J. Complexity 20, No. 6, 858--875 (2004; Zbl 1064.41018)]. Here the authors determine asymptotic orders of linear probabilistic \((N, \delta)\)-widths \(\lambda_{N,\delta}\) and linear \(p\)-average widths \(\lambda_N^{(a)}\) of the set \(MW_2^r({\mathbb T}^d)\) of multivariate functions with bounded mixed derivative equipped with the Gaussian measure \(\mu\) in \(L_q({\mathbb T}^d)\). If \(1<q<\infty\) and \(r=(r_1, \ldots , r_d)\), \(1/2<r_1=\ldots =r_\nu<r_{\nu+1}\leq \ldots \leq r_d\), then \[ \lambda_N^{(a)}\left ( MW_2^r({\mathbb T}^d), \mu , L_q({\mathbb T}^d) \right )\asymp (N^{-1}\ln^{\nu-1}N)^{r_1+(\rho-1/2)}\left (\ln^{(\nu-1)/2} N \right ), \] where \(\rho>1\) depends only on the eigenvalues of the correlation operator of the measure \(\mu\). A similar equivalence is established for \((N, \delta)\)-widths.
Mathematics Subject Classification ID: 41A46
Mathematics Subject Classification ID: 41A63
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 2136635
zbMATH Keywords: average widhs, Sobolev spaces
zbMATH Keywords: bounded mixed derivative
reviewed by: Yuly Makovoz
MaRDI profile type: Publication
full work available at URL: j.jat.2004.10.012
OpenAlex ID: W2064440937
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DOI: 10.1016/J.JAT.2004.10.012
description: scientific article; zbMATH DE number 2136635