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Wavelets with crystal symmetry shifts. (English) Zbl 1231.42040

Summary: A generalization of multi-dimensional wavelet theory is introduced in which the usual lattice of translational shifts is replaced by a discrete subgroup of the group of affine, area preserving, transformations of Euclidean space. The dilation matrix must now be compatible with the group of shifts. An existence theorem for a multiwavelet in the presence of a multiresolution analysis is established and examples are given to illustrate the theory with two dimensional crystal symmetry groups as shifts.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
20H15 Other geometric groups, including crystallographic groups
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