Crystallographic Haar-Type Composite Dilation Wavelets

Abstract

AbstractAn (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.KeywordsScaling FunctionFundamental RegionInvertible MatriceCrystallographic GroupParseval FrameThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.