``Push-the-Error'' algorithm for nonlinear \(n\)-term approximation
type: Item
instance of: scholarly article
title: ``Push-the-Error'' algorithm for nonlinear \(n\)-term approximation
zbMATH Open document ID: 1106.41023
author: Wolfgang Dahmen
author: Pencho Petrushev
published in: Constructive Approximation
publication date: 2006-11-17T00:00:00Z
review text: Let \(\Omega\) be a compact domain in \({\mathbb R}^d\) and let \(V_m\), \(m=0,1,2,\ldots\), be subspaces of \(C(\Omega)\) such that \(V_0 \subset V_1 \subset \ldots\) and \(\overline {{\cup V_m}}=C(\Omega)\). It is assumed that for every \(m\) there is a finite set \(\Theta_m \subset \{0,1, \ldots \}\) so that \(V_m\) is spanned by a basis \(\{\phi_\theta \}_ {\theta \in \Theta_m}\) and a given \(f \in C(\Omega)\) can be represented in the form \[ f=\sum_{m=0}^\infty \sum_{ \theta \in \Theta_m} b_\theta \phi_\theta. \] The authors consider an algorithm that produces for any target accuracy an approximation to \(f\) of the form \(\sum d_\theta \phi_\theta\) with a small number of non-zero coefficients \(d_\theta\). Compared with the previous work in the field, the paper offers a more general setting for multiresolution analysis. The algorithm is especially designed to deal with the \(L_\infty\) approximation. The main conceptually new result is the quasi-subadditivity, with respect to \(f\), of the functional \(N(\varepsilon)\) counting the number of terms needed to achieve accuracy \(\epsilon\).
Mathematics Subject Classification ID: 41A46
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 41A17
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 65D15
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 5073584
zbMATH Keywords: multilevel expansions
zbMATH Keywords: wavelet bases
zbMATH Keywords: \(n\)-term approximation
reviewed by: Yuly Makovoz
MaRDI profile type: Publication
full work available at URL: s00365-005-0607-8
OpenAlex ID: W2107700451
DOI: 10.1007/S00365-005-0607-8
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description: scientific article; zbMATH DE number 5073584