Generalized multi-resolution analyses and a construction procedure for all wavelet sets in \(\mathbb{R}^n\)
type: Item
instance of: scholarly article
title: Generalized multi-resolution analyses and a construction procedure for all wavelet sets in \(\mathbb{R}^n\)
zbMATH Open document ID: 0972.42021
DOI: 10.1007/BF01257191
author: H. Medina
author: Kathy D. Merrill
author: Lawrence W. Baggett
published in: The Journal of Fourier Analysis and Applications
publication date: 2000-09-04T00:00:00Z
full work available at URL: https://eudml.org/doc/59623
review text: Let \(A\) be a real integer \(n\times n\) matrix, for which all eigenvalues have absolute value greater than one (i.e., an expansive matrix) and define a unitary operator on \(L^2(R^n)\) by \([\delta_Af](x)= |\det A |^{1/2}f(Ax)\). A set \(W\subseteq R^n\) is a wavelet set (with respect to \(A\)) if the function \(\psi\) defined by \(\chi_W=\hat{\psi}\) is a wavelet (i.e., if \(\{|\det A|^{m/2} \psi(A^m \cdot -k)\}_{m\in Z, k\in Z^n}\) is an orthonormal basis for \(L^2(R^n)\)). A generalized multiresolution analysis (GMRA) associated to \(A\) is an increasing family of closed subspaces \(\{V_j\}\) of \(L^2(R)\) for which (i) \(V_{j+1}=\delta_A V_j\) for all \(j\), (ii) \(\cap V_j = \{0\}\) and \(\overline {\cup V_j}= L^2(R^n)\), (iii) \(V_0\) is invariant under the action of the translation operators \((\gamma_kf)(x)= f(x+k), k\in Z^n\). The relationship between GMRA's and multiwavelets is explored, and as a consequence the authors are able to construct all wavelet sets in \(R^n\), not only the ones arising from a multiresolution analysis. Several new wavelet sets are constructed.
Mathematics Subject Classification ID: 42C40
zbMATH DE Number: 1441219
zbMATH Keywords: wavelet
zbMATH Keywords: wavelet set
zbMATH Keywords: multiresolution analysis
zbMATH Keywords: unitary representation
MaRDI profile type: Publication
cites work: General existence theorems for orthonormal wavelets, an abstract approach
cites work: Wandering vectors for unitary systems and orthogonal wavelets
cites work: Wavelet sets in \(\mathbb{R}^ n\)
cites work: Q4381495
cites work: Q5808569
cites work: The spectral theorem
cites work: Q4111791
cites work: Q4334625
OpenAlex ID: W2003176204
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description: scientific article; zbMATH DE number 1441219