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Search: All articles in the CJM digital archive with keyword higher rank

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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 squares
Categories:42C40, 42A65, 42A16, 43A65

2. CJM 2009 (vol 61 pp. 1239)

Davidson, Kenneth R.; Yang, Dilian
Periodicity in Rank 2 Graph Algebras
Kumjian and Pask introduced an aperiodicity condition for higher rank graphs. We present a detailed analysis of when this occurs in certain rank 2 graphs. When the algebra is aperiodic, we give another proof of the simplicity of $\mathrm{C}^*(\mathbb{F}^+_{\theta})$. The periodic $\mathrm{C}^*$-algebras are characterized, and it is shown that $\mathrm{C}^*(\mathbb{F}^+_{\theta}) \simeq \mathrm{C}(\mathbb{T})\otimes\mathfrak{A}$ where $\mathfrak{A}$ is a simple $\mathrm{C}^*$-algebra.

Keywords:higher rank graph, aperiodicity condition, simple $\mathrm{C}^*$-algebra, expectation
Categories:47L55, 47L30, 47L75, 46L05

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