More regular wavelets
type: Item
instance of: scholarly article
title: More regular wavelets
zbMATH Open document ID: 0912.42022
published in: Applied and Computational Harmonic Analysis
publication date: 1999-04-27T00:00:00Z
review text: If \(\psi\) is a wavelet with Fourier transform \(\widehat{\psi}\), then its Sobolev regularity exponent is \(\sigma=\sup\{s: | \xi| ^s\widehat{\psi}(\xi)\in L^2\}\). Let \(\phi\) be the scaling function in the corresponding multiresolution analysis. The Daubechies orthogonal compactly supported wavelets with support \(2N-1\) have regularity exponent \(\sigma_N\) with \(\lim_{N\to\infty}(\sigma_N/2N)=1/2-\ln 3/4\ln 2\). The proof of this result is revised, and then adapted to show that there is a class of wavelets introduced by Matzinger that have the same support as Daubechies wavelets, but have a better regularity exponent. The ``equality'' sign in the above limit can be replaced by a ``greater than'' sign. Thus the regularity/complexity ratio is improved. The Daubechies wavelets \(\psi_N\) have order of approximation \(N-1\), which is expressed by the fact that the scaling filter \(m_0(\xi)\) has a zero of order \(N-1\) in \(\pi\). In the case of Matzinger wavelets \(\psi_{L,M}\) this becomes a zero of order \(L\) in \(\pi\) and a zero of order \(2M\) in \(2\pi/3\). The Matzinger \(\psi_{L,M}\) should be compared with the Daubechies \(\psi_{L+2M}\) since they have the same filter length.
Mathematics Subject Classification ID: 42C40
zbMATH DE Number: 1124819
zbMATH Keywords: compactly supported wavelets
zbMATH Keywords: Matzinger wavelets
zbMATH Keywords: Sobolev regularity exponent
zbMATH Keywords: scaling function
zbMATH Keywords: multiresolution analysis
zbMATH Keywords: Daubechies wavelets
author: Pierre Gilles Lemarié-Rieusset
author: Ezzeddine Zahrouni
reviewed by: Adhemar Bultheel
MaRDI profile type: Publication
OpenAlex ID: W1979828043
DOI: 10.1006/ACHA.1997.0225
description: scientific article; zbMATH DE number 1124819