Continuous algorithms in \(n\)-term approximation and nonlinear widths
type: Item
instance of: scholarly article
title: Continuous algorithms in \(n\)-term approximation and nonlinear widths
zbMATH Open document ID: 0951.41011
author: Dinh Dũng
published in: Journal of Approximation Theory
publication date: 2000-12-18T00:00:00Z
review text: Let \(X \) be a quasi-normed linear space and \(\Phi = \{\phi_k\}_{k=1}^{\infty} \) a family of elements in \(X\). If the family \(\Phi \) is bounded, i.e., \(\|\phi_k\|\leq C\), \(k=1,2,\dots\), and the span of \(\Phi \) is dense in \(X\), then \(\Phi \) is called a dictionary. Denote by \(M_n(\Phi)\) the set of all linear combinations \(\phi =\sum_{k\in Q}a_k\phi _k\), where \(Q\) is a set of natural numbers with \(|Q|=n\). If \(W\) is a subset \(X\) then the quantity \(\sigma_n(W,\Phi ,X) = \sup_{f\in W}\inf_{\phi \in M_n(\Phi)} \|f-\phi \|\) is called the \(n\)-term approximation of \(W\) by the family \(\Phi \). In this paper the author considers optimal continuous algorithms in \(n\)-term approximation based on various non-linear \(n\)-widths, and the \(n\)-term approximation \(\sigma_n(W,V,X)\) by the dictionary \(V\) formed from the integer translates of the mixed dyadic scales of the tensor product multivariate de la Vallée Poussin kernel, for the unit ball of Sobolev and Besov spaces of functions with a common mixed smoothness. The asymptotic orders of these quantities are given. These orders are achieved by a continuous algorithm of \(n\)-term approximation by \(V\), which is explicitly constructed.
Mathematics Subject Classification ID: 41A46
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 41-02
zbMATH DE Number: 1423442
zbMATH Keywords: \(n\)-term approximation
zbMATH Keywords: \(n\)-widths
zbMATH Keywords: Besov spaces
zbMATH Keywords: Sobolev spaces
zbMATH Keywords: Vallée Poussin kernel
reviewed by: Saulius Norvidas
MaRDI profile type: Publication
full work available at URL: jath.1999.3399
OpenAlex ID: W2065058944
cites work: Q4215356
cites work: Optimal nonlinear approximation
cites work: Wavelet compression and nonlinear \(n\)-widths
cites work: Q3772674
cites work: Q4694392
cites work: On nonlinear \(n\)-widths and \(n\)-term approximation
cites work: On best continuous methods in \(n\)-term approximation
cites work: On nonlinear 𝑛-widths
cites work: On best \(m\)-term approximations and the entropy of sets in the space \(L^ 1\)
cites work: s-numbers in information-based complexity
cites work: Q4064526
cites work: On the value of partial information for learning from examples
cites work: Q3027578
cites work: Nonlinear Kolmogorov widths
cites work: Greedy algorithms with regard to multivariate systems with special structure
DOI: 10.1006/JATH.1999.3399
description: scientific article; zbMATH DE number 1423442