Generalized wavelets and compactly supported scaling functions
type: Item
instance of: scholarly article
title: Generalized wavelets and compactly supported scaling functions
zbMATH Open document ID: 0784.42025
DOI: 10.4171/RMI/140
published in: Revista Matemática Iberoamericana
publication date: 1993-08-31T00:00:00Z
full work available at URL: https://eudml.org/doc/39442
review text: This paper solves two old problems in wavelet theory concerning the relation between multiresolution analyses and wavelet bases in dimension 1 with a general dilation factor \(A\), \(A \in Z\), \(A \geq 2\). A ``multiresolution analysis'' (MRA in short) with respect to a dilation \(A\) is an increasing sequence \((V_ j)_{j \in Z}\) of closed subspaces of \(L^ 2(R)\) with the following properties: (1) \(\bigcup_{j \in Z}V_ j\) is dense in \(L^ 2(R)\) and \(\bigcap_{j \in Z}V_ j=\{0\}\), (2) \(f(x) \in V_ j \to f(Ax) \in V_{j+1}\) for all \(j \in Z\), (3) \(V_ 0\) possesses an orthonormal basis of the form \(\bigl\{ \varphi(x- k)\), \(k \in Z \bigr\}\). \(\varphi\) is called the scaling function of the MRA. A ``wavelet basis'' generated by the wavelets \(\psi_ 1,\ldots, \psi_{A-1}\) is an orthonormal basis of the form \(\bigl\{ A^{j/2} \psi_ r (A^ jx-k),\;j,k \in Z,\;r=1,\ldots,A-1\bigr\}\). For the construction of a wavelet basis one considers the orthogonal complement \(W_ 0\) of \(V_ 0\) in \(V_ 1\) and shows that there exist wavelets \(\psi_ 1,\ldots,\psi_{A-1}\), such that the integer translates of the \(\psi_ r\)'s form an orthonormal basis of \(W_ 0\). It is known that these wavelets can be chosen to have the same smoothness and decay properties as the scaling function \(\varphi\), but for \(A>2\) no construction of compactly supported wavelets had been obtained. All interesting wavelet bases have been derived from a MRA following this strategy. Therefore it had been conjectured since the beginnings of wavelet theory that this is the only possible construction for wavelet basis with sufficient smoothness and decay properties. This paper solves both problems and provides several versions of the relation between MRA's and wavelet bases. Theorem 1: If \((V_ j)_{j \in Z}\) is a MRA with dilation factor \(A \geq 2\), \(A \in Z\), and with a compactly supported scaling function \(\varphi\), then there exist \(A-1\) compactly supported wavelets \(\psi_ r\) so that \(\…
Mathematics Subject Classification ID: 42C40
zbMATH DE Number: 404302
zbMATH Keywords: translation invariant subspaces
zbMATH Keywords: biorthogonal basis
zbMATH Keywords: multiresolution analysis
zbMATH Keywords: dilation factor
zbMATH Keywords: wavelet basis
zbMATH Keywords: compactly supported wavelets
zbMATH Keywords: compactly supported scaling function
author: Pierre Gilles Lemarié-Rieusset
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 404302