Crystallographic Haar-Type Composite Dilation Wavelets

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


An (a, B, Γ) composite dilation wavelet system is a collection of functions generating an orthonormal basis for L 2( 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( 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 n and the theory is applied to the 17 crystallographic groups acting on 2 where 11 are shown to admit such Haar-type systems.


Scaling Function Fundamental Region Invertible Matrice Crystallographic Group Parseval Frame 


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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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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsGrinnell CollegeGrinnellUSA

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