Linear average and stochastic \(n\)-widths of Besov embeddings on Lipschitz domains
type: Item
instance of: scholarly article
title: Linear average and stochastic \(n\)-widths of Besov embeddings on Lipschitz domains
zbMATH Open document ID: 1185.41023
published in: Journal of Approximation Theory
publication date: 2009-11-26T00:00:00Z
review text: For a given bounded linear operator \(T:X\to Y\), denote by \(\Delta_{\mu}(T_n)=\| T(\cdot)-T_n(\cdot)\|_{L^2(\mu,Y)}\) the average error of \(T\) approximated by \(T_n:X\to Y\) with \(\dim T_n(X)\leq n\), with respect to a probability Borel measure \(\mu\) on the unit ball of \(X\), and \(a_n^{\text{avg}}(T)=\inf_{T_n,\mu} \Delta_{\mu}(T_n)\) the linear \(n\)-width average of \(T\). Similarly, the linear stochastic \(n\)-width is defined as \(a_n^{\text{ran}}(T)=\inf_{T^\omega_n,\mu} \Delta_{\max}(T^\omega_n)\) with \(\Delta_{\max}(T^\omega_n)=\sup_{\| f\|_X\leq 1}(E(\| T(f)-T_n^\omega(f)\|_Y^2))^{1/2}\) if \(\dim T^\omega_n(X)\leq n\), and \(T^\omega_n\) is randomly chosen. The authors consider the compact embeddings \(I:B_{q_0}^{s+t}(L^{p_0}(\Omega))\to B_{q_1}^{s}(L^{p_1}(\Omega))\) of Besov spaces into a bounded Lipschitz domain \(\Omega\subset{\mathbb R}^d\), with \(t>d(1/p_0-1/p_1)_+\). They show that \(a_n^{\text{avg}}(I)\) and \(a_n^{\text{ran}}(I)\) behave like powers \(n^{-\alpha(t)}\), where \(\alpha(t)\) is a linear function of \(t\). For instance, if \(1\leq p_0\leq p_1\leq 2\), then \(\alpha(t)=t/d-(1/p_0-1/p_1)\). The proof is based on a characterization of Besov spaces by means of wavelet coefficients.
Mathematics Subject Classification ID: 41A45
Mathematics Subject Classification ID: 42A61
zbMATH DE Number: 5639029
zbMATH Keywords: average widths
zbMATH Keywords: stochastic widths
zbMATH Keywords: Besov space
zbMATH Keywords: Lipschitz domain
zbMATH Keywords: wavelets coefficients
author: Gensun Fang
author: Lixin Qian
reviewed by: Joan Cerdà
MaRDI profile type: Publication
full work available at URL: j.jat.2008.05.004
OpenAlex ID: W2093721459
cites work: Q3026133
cites work: Q3995208
cites work: Probabilistic and average widths of multivariate Sobolev spaces with mixed derivative equipped with the Gaussian measure
cites work: Q4434436
cites work: Besov regularity for elliptic boundary value problems
cites work: Optimal approximation of elliptic problems by linear and nonlinear mappings. II
cites work: Wavelet bases and entropy numbers in weighted function spaces
cites work: Lower bounds for the complexity of Monte Carlo function approximation
cites work: Entropy numbers of embeddings of weighted Besov spaces
cites work: Average \(n\)-widths of the Wiener space in the \(L_ \infty\)-norm
cites work: Linear widths of function spaces equipped with the Gaussian measure
cites work: About widths of Wiener space in the \(L_ q\)-norm
cites work: Widths and distributions of values of the approximation functional on the Sobolev spaces with measure
cites work: Q4081833
cites work: Q4695859
cites work: Probabilistic and average linear width in \(L_ \infty\)-norm with respect to \(r\)-fold Wiener measure
cites work: Random approximation of Sobolev embeddings
cites work: A \(q\)-analogue of the Coxeter complex
cites work: s-numbers in information-based complexity
cites work: Q3135829
cites work: Q5577906
cites work: Optimal linear randomized methods for linear operators in Hilbert spaces
cites work: Q4183861
cites work: Q4722695
cites work: Q3344608
cites work: Average-case analysis of numerical problems
cites work: Q4893982
cites work: Q3993279
cites work: Q3336200
cites work: Entropy Numbers and Approximation Numbers in Function Spaces, II
cites work: Q4357054
cites work: Q4828793
DOI: 10.1016/J.JAT.2008.05.004
description: scientific article; zbMATH DE number 5639029