An adaptive compression algorithm in Besov spaces
type: Item
instance of: scholarly article
title: An adaptive compression algorithm in Besov spaces
zbMATH Open document ID: 1004.41006
author: L. Birgé
author: Pascal Massart
published in: Constructive Approximation
publication date: 2003-02-11T00:00:00Z
review text: Let \(\overline F\) be an \(\overline M\)-dimensional linear space of functions on \([0,1]^d\) and let \(F\) be an \(M\)-dimensional linear space of compactly supported functions in \(L_\infty(\mathbb{R}^d)\). Define, for \(j\geq 0\) an integer and \({\mathbf k}\in \mathbb{Z}^d\), \[ F_{j,{\mathbf k}}= \{g(2^j x-{\mathbf k}): g\in F\}. \] The paper presents a compression algorithm for approximating a function \(f\in L_1([0, 1]^d)\) which has on \([0,1]^d\) an expansion \[ f=\overline f+ \sum^\infty_{j= 0} \sum_{{\mathbf k}\in\Lambda(j)} f_{j,{\mathbf k}},\tag{\(*\)} \] where \(\Lambda(j)\subseteq \mathbb{Z}^d\) and \(2^{jd}\leq |\Lambda(j)|\leq \lambda 2^{jd}\) for all \(j\geq 0\), and in which \(\overline f\in\overline F\) and each \(f_{j,{\mathbf k}}\in F_{j,{\mathbf k}}\). The algorithm \({\mathbf A}({\mathcal L},{\mathbf J})\) depends upon a slowly varying increasing function \({\mathcal L}:(0, 1]\to (0,1]\) satisfying prescribed conditions and an integer \(J\geq 0\). It is defined as follows. Let \(K_j= [{\mathcal L}(2^{j-j}) 2^{d(J- j)}|\Lambda(j)|]\) for \(j\geq J\). For \(0\leq j< J\) define \(\Lambda'(j)= \Lambda(j)\) and for \(j\geq J\) let \(\Lambda'(j)\) be a subset of \(\Lambda(j)\) such that \(|\Lambda'(j)|= K_j\) and (the crux of the algorithm) \[ \|f_{j,{\mathbf k}'}\|_2\leq\|f_{j,{\mathbf k}}\|_2\text{ for all }{\mathbf k}\in \Lambda'(j),\;{\mathbf k}'\not\in \Lambda'(j). \] Take as an approximation to \(f\) the finite sum \[ \widetilde f_J=\overline f+ \sum^\infty_{j=0} \sum_{{\mathbf k}\in \Lambda'(j)} f_{j,{\mathbf k}}. \] The paper is concerned in particular with three types of expansions of the form \((*)\) and they are discussed, for simplicity, in the case \(d=1\): (1) \(f\in L_2([0, 1])\) and the expansion \((*)\) is derived from orthogonal projections, to the spaces of polynomials of degree \(\leq r-1\), of the restrictions of \(f\) to dyadic subintervals of \([0,1]\); (2) orthonormal wavelet expansions with an \((r-1)\)-regular mother wave…
Mathematics Subject Classification ID: 41A25
Mathematics Subject Classification ID: 41A46
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 41A30
zbMATH DE Number: 1419732
zbMATH Keywords: approximation
zbMATH Keywords: algorithm
zbMATH Keywords: signal compression
zbMATH Keywords: threshold
zbMATH Keywords: Besov space
zbMATH Keywords: piecewise polynomial
zbMATH Keywords: spline
zbMATH Keywords: wavelet
zbMATH Keywords: entropy
reviewed by: Aldric L. Brown
describes a project that uses: Wavelet Toolbox
MaRDI profile type: Publication
full work available at URL: s003659910001
OpenAlex ID: W2164782566
DOI: 10.1007/S003659910001
description: scientific article; zbMATH DE number 1419732