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Higher Rank Wavelets

Published online by Cambridge University Press:  20 November 2018

Sean Olphert
Affiliation:
Department of Mathematics & Statistics, Lancaster University, Lancaster LA1 4YF, UK email: s.olphert@lancaster.ac.uks.power@lancaster.ac.uk
Stephen C. Power
Affiliation:
Department of Mathematics & Statistics, Lancaster University, Lancaster LA1 4YF, UK email: s.olphert@lancaster.ac.uks.power@lancaster.ac.uk
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Abstract

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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 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.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2011

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