Crystallographic Haar wavelets
type: Item
instance of: scholarly article
title: Crystallographic Haar wavelets
zbMATH Open document ID: 1270.42032
author: Alfredo L. González
author: María del Carmen Moure
published in: The Journal of Fourier Analysis and Applications
publication date: 2012-01-10T00:00:00Z
review text: Let \(\Gamma\) be a \(d\)-dimensional crystallographic group and let \(a:\,{\mathbb R}^d \to {\mathbb R}^d\) be an expanding affine map. By definition, \((\Gamma,a)\)-crystallographic multiwavelets form a finite set of functions \(\{\psi^1,\ldots, \psi^L\}\), which generate an orthonormal basis, a Riesz basis or a Parseval frame for \(L^1({\mathbb R}^d)\) under the action of \(\Gamma\) and powers of \(a\), taking the place of translations and dilations in classical wavelets. This paper on crystallographic Haar-type multiwavelets is based on self-replicant tilings (see \textit{M. del C. Moure} [Discrete Comput. Geom. 42, No. 4, 722--739 (2009; Zbl 1186.28007)]), reptiles associated to crystallographic groups (see \textit{G. Gelbrich} [Geom. Dedicata 51, No. 3, 235--256 (1994; Zbl 0807.05022)]) and characterizations of scaling functions for Haar-type multiresolution analyses (see [\textit{K. Gröchenig} and \textit{W.R. Madych}, IEEE Trans. Inf. Theory 38, 556--568 (1994)]). The authors characterize scaling function vectors for Haar-type crystallographic \((\Gamma,a)\)-multiresolution analyses. Then corresponding \((\Gamma,a)\)-crystallographic multiwavelets of Haar type are constructed. Examples are presented in the cases \(d=2\) and \(d=3\).
reviewed by: Manfred Tasche
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 52C22
Mathematics Subject Classification ID: 20H15
zbMATH DE Number: 5996269
zbMATH Keywords: crystallographic multiwavelet
zbMATH Keywords: Haar-type multiwavelets
zbMATH Keywords: crystallographic group
zbMATH Keywords: expanding affine map
zbMATH Keywords: crystallographic multiresolution analysis
zbMATH Keywords: scaling function vectors
zbMATH Keywords: tilings
zbMATH Keywords: reptiles
MaRDI profile type: Publication
full work available at URL: s00041-011-9175-4
OpenAlex ID: W2007891200
cites work: Wavelets on irregular grids with arbitrary dilation matrices and frame atoms for \(L^2(\mathbb R{^d})\)
cites work: Q4858541
cites work: General existence theorems for orthonormal wavelets, an abstract approach
cites work: Matricial filters and crystallographic composite dilation wavelets
cites work: Crystallographic Haar-Type Composite Dilation Wavelets
cites work: Ridgelets: a key to higher-dimensional intermittency?
cites work: New tight frames of curvelets and optimal representations of objects with piecewise C2 singularities
cites work: Haar-type multiwavelet bases and self-affine multi-tiles
cites work: Crystallographic reptiles
cites work: Multiresolution analysis. Haar bases, and self-similar tilings of R/sup n/
cites work: Self-affine tilings with several tiles. I
cites work: Q3735714
cites work: Wavelets with composite dilations
cites work: Wavelets with composite dilations and their MRA properties
cites work: Q5493732
cites work: Crystallography and Cohomology of Groups
cites work: Q3731958
cites work: Some simple Haar-type wavelets in higher dimensions
cites work: Self-affine tiles in \(\mathbb{R}^n\)
cites work: Q3256335
cites work: Wavelets with crystal symmetry shifts
cites work: A framework for the construction of self-replicating tilings
cites work: Graph-directed reptiles of \(\mathbb R^2\)
cites work: Periodicity, Quasiperiodicity, and Bieberbach's Theorem on Crystallographic Groups
DOI: 10.1007/S00041-011-9175-4
description: scientific article; zbMATH DE number 5996269