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Search: All articles in the CJM digital archive with keyword multiresolution

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1. CJM 2011 (vol 63 pp. 689)

Olphert, Sean; Power, Stephen C.
 Higher Rank Wavelets 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 squaresCategories:42C40, 42A65, 42A16, 43A65

2. CJM 2008 (vol 60 pp. 334)

Curry, Eva
 Low-Pass Filters and Scaling Functions for Multivariable Wavelets We show that a characterization of scaling functions for multiresolution analyses given by Hern\'{a}ndez and Weiss and that a characterization of low-pass filters given by Gundy both hold for multivariable multiresolution analyses. Keywords:multivariable multiresolution analysis, low-pass filter, scaling functionCategories:42C40, 60G35

3. CJM 2002 (vol 54 pp. 634)

Weber, Eric
 Frames and Single Wavelets for Unitary Groups We consider a unitary representation of a discrete countable abelian group on a separable Hilbert space which is associated to a cyclic generalized frame multiresolution analysis. We extend Robertson's theorem to apply to frames generated by the action of the group. Within this setup we use Stone's theorem and the theory of projection valued measures to analyze wandering frame collections. This yields a functional analytic method of constructing a wavelet from a generalized frame multi\-resolution analysis in terms of the frame scaling vectors. We then explicitly apply our results to the action of the integers given by translations on $L^2({\mathbb R})$. Keywords:wavelet, multiresolution analysis, unitary group representation, frameCategories:42C40, 43A25, 42C15, 46N99