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# Higher Rank Wavelets

Published:2011-02-25
Printed: Jun 2011
• Sean Olphert,
Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, UK
• Stephen C. Power,
Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, UK
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## Abstract

A theory of higher rank multiresolution analysis is given in the setting of abelian multiscalings. This theory enables the construction, from a higher rank MRA, of finite wavelet sets whose multidilations have translates forming an orthonormal basis in $L^2(\mathbb R^d)$. While tensor products of uniscaled MRAs provide simple examples we construct many nonseparable higher rank wavelets. In particular we construct \emph{Latin square wavelets} as rank~$2$ variants of Haar wavelets. Also we construct nonseparable scaling functions for rank $2$ variants of Meyer wavelet scaling functions, and we construct the associated nonseparable wavelets with compactly supported Fourier transforms. On the other hand we show that compactly supported scaling functions for biscaled MRAs are necessarily separable.
 Keywords: wavelet, multi-scaling, higher rank, multiresolution, Latin squares
 MSC Classifications: 42C40 - Wavelets and other special systems 42A65 - Completeness of sets of functions 42A16 - Fourier coefficients, Fourier series of functions with special properties, special Fourier series {For automorphic theory, see mainly 11F30} 43A65 - Representations of groups, semigroups, etc. [See also 22A10, 22A20, 22Dxx, 22E45]

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