Crystallographic Haar-Type Composite Dilation Wavelets

  • Jeffrey D. Blanchard
  • Kyle R. Steffen
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 
These 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.


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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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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsGrinnell CollegeGrinnellUSA

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