Crystallographic Haar-Type Composite Dilation Wavelets
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.
KeywordsScaling 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.
- 2.J. D. Blanchard. Minimally supported frequency composite dilation wavelets. J. Fourier Anal. App. Online First, 2009, DOI:10.1007/s00041-009-9080-2.Google Scholar
- 4.J. D. Blanchard and I. A. Krishtal. Matricial filters and crystallographic composite dilation wavelets. submitted, 2009, http://www.math.grin.edu/~blanchaj/Research/MFCCDW_BlKr.pdf
- 8.I. Daubechies. Ten lectures on wavelets, volume 61 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992.Google Scholar
- 10.K. Gröchenig and W. R. Madych. Multiresolution analysis, Haar bases, and self-similar tilings of R n. IEEE Trans. Inform. Theory, 38(2, part 2):556–568, 1992.Google Scholar
- 12.K. Guo, G. Kutyniok, and D. Labate. Sparse multidimensional representations using anisotropic dilation and shear operators. In Wavelets and splines: Athens 2005, Mod. Methods Math., pages 189–201. Nashboro Press, Brentwood, TN, 2006.Google Scholar
- 15.K. Guo, D. Labate, W. Lim, G. Weiss, and E. N. Wilson. The theory of wavelets with composite dilations. In Harmonic analysis and applications, Appl. Numer. Harmon. Anal., pages 231–250. Birkhäuser Boston, Boston, MA, 2006.Google Scholar
- 19.R. Houska. The nonexistence of shearlet-like scaling multifunctions that satisfy certain minimally desirable properties and characterizations of the reproducing properties of the integer lattice translations of a countable collection of square integrable functions. Ph.D. dissertation, 2009. Washington University in St. Louis.Google Scholar
- 22.J. MacArthur. Compatible dilations and wavelets for the wallpaper groups. preprint, 2009.Google Scholar
- 23.J. MacArthur and K. Taylor. Wavelets from crystal symmetries. preprint, 2009.Google Scholar