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- W66786174 abstract "An (a, B, Γ) composite dilation wavelet system is a collection of functions generating an orthonormal basis for L 2(ℝ n ) under the actions of translations from a full rank lattice, Γ, dilations by elements of B, a subgroup of the invertible n ×n matrices, and dilations by integer powers of an expanding matrix a. A Haar-type composite dilation wavelet system has generating functions which are linear combinations of characteristic functions. Krishtal, Robinson, Weiss, and Wilson introduced three examples of Haar-type (a, B, Γ) composite dilation wavelet systems for L 2(ℝ 2) under the assumption that B is a finite group which fixes the lattice Γ. We establish that for any Haar-type (a, B, Γ) composite dilation wavelet, if B fixes Γ, known as the crystallographic condition, B is necessarily a finite group. Under the crystallographic condition, we establish sufficient conditions on (a, B, Γ) for the existence of a Haar-type (a, B, Γ) composite dilation wavelet. An example is constructed in ℝ n and the theory is applied to the 17 crystallographic groups acting on ℝ 2 where 11 are shown to admit such Haar-type systems." @default.
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- W66786174 date "2011-01-01" @default.
- W66786174 modified "2023-09-27" @default.
- W66786174 title "Crystallographic Haar-Type Composite Dilation Wavelets" @default.
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- W66786174 doi "https://doi.org/10.1007/978-0-8176-8095-4_5" @default.
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