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Crystallographic Haar-Type Composite Dilation Wavelets

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Book cover Wavelets and Multiscale Analysis

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

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.

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Acknowledgements

This work was partially supported by the University of Utah VIGRE program funded by NSF DMS grant number 0602219. JDB was partially funded as a VIGRE research assistant professor. KRS was funded through a year-long REU. The authors thank Guido Weiss, Ed Wilson, Ilya Krishtal, Keith Taylor, and Josh MacArthur for our rewarding conversations regarding CHCDW.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Grinnell College, Grinnell, Iowa, USA

    Jeffrey D. Blanchard

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  1. Jeffrey D. Blanchard
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  2. Kyle R. Steffen
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Correspondence to Jeffrey D. Blanchard .

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  1. , Department of Mathematical Sciences, DePaul University, North Kenmore Avenue 2320, Chicago, 60614-3250, Illinois, USA

    Jonathan Cohen

  2. , Department of Mathematical Sciences, DePaul University, North Kenmore Avenue 2320, Chicago, 60614-3250, Illinois, USA

    Ahmed I. Zayed

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Blanchard, J.D., Steffen, K.R. (2011). Crystallographic Haar-Type Composite Dilation Wavelets. In: Cohen, J., Zayed, A. (eds) Wavelets and Multiscale Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-8095-4_5

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