Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Higher Rank Wavelets
AU - Olphert, Sean
AU - Power, Stephen
PY - 2011
Y1 - 2011
N2 - 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 . 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.
AB - 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 . 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.
KW - wavelet
KW - multi-scaling
KW - higher rank
KW - multiresolution
KW - Latin squares
U2 - 10.4153/CJM-2011-012-1
DO - 10.4153/CJM-2011-012-1
M3 - Journal article
VL - 63
SP - 689
EP - 720
JO - Canadian Journal of Mathematics
JF - Canadian Journal of Mathematics
SN - 0008-414X
ER -