Composite Dilations and Crystallographic Groups

  • Kathy D. Merrill
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


This chapter explores GMRAs in the familiar Hilbert space of \(L^2(\mathbb R^N)\), but with a non-abelian group Γ of “translations” that properly contains the integer lattice. Guo, Labate, Lim, Weiss and Wilson’s theory of composite dilations is included, as well as GMRAs and wavelets for the crystallographic groups.


  1. 1.
    Auslander, L.: An account of the theory of crystallographic groups. Proc. Am. Math. Soc. 16, 1230–1236 (1965)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Blanchard, J., Krishtal, I.: Matricial filters and crystallographic composite dilation wavelets. Math. Comput. 81, 905–922 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Blanchard, J., Steffen, K.: Crystallographic Haar-type composite dilation wavelets. In: Cohen, J., Zyed, A.I. (eds.) Wavelets and Multiscale Analysis: Theory and Applications, pp 83–108. Birkhäuser, Boston (2011)CrossRefGoogle Scholar
  4. 4.
    Federov, E.: Elements of the Theory of Figures. Imp. Acad. Sci., St. Petersburg (1885)Google Scholar
  5. 5.
    Federov, E.: Symmetry in the plane. Zapiski Rus. Mineralog. Obščestva, Ser. 2 28, 345–390 (1891)Google Scholar
  6. 6.
    Flaherty, T., Wang, Y.: Haar-type multiwavelet bases and self-affine multi-tiles. Asian J. Math. 3, 387–400 (1999)MathSciNetCrossRefGoogle Scholar
  7. 7.
    González, A.L., Moure, M.C.: Crystallographic Haar wavelets. J. Fourier Anal. Appl. 17, 1119–1137 (2011)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Groöchenig, K., Madych, W.R.: Multiresolution analysis, Haar bases, and self-similar tilings of \(\mathbb R^n\). IEEE Trans. Inf. Theory 38, 556–568 (1992)Google Scholar
  9. 9.
    Groöchenig, K., Haas, A., Raugi, A.: Self-affine tilings with several tiles. Appl. Comput. Harmon. Anal. 7, 211–238 (1999)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Guo, K., Labate, D., Lim, W.-Q, Weiss, G., Wilson, E.: Wavelets with composite dilations. Electron. Res. Announc. Am. Math. Soc. 10, 78–87 (2004)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Guo, K., Labate, D., Lim, W.-Q, Weiss, G., Wilson, E.: The theory of wavelets with composite dilations. In: Heil, C. (ed.) Harmonic Analysis and Applications, pp. 231–250. Birkhäuser, Boston (2006)CrossRefGoogle Scholar
  12. 12.
    Guo, K., Labate, D., Lim, W.-Q, Weiss, G., Wilson, E.: Wavelets with composite dilations and their MRA properties. Appl. Comput. Harmon. Anal. 20, 202–236 (2006)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Krishtal, I., Robinson, B., Weiss, G., Wilson, E.: Some simple Haar-type wavelets in higher dimensions. J. Geom. Anal. 17, 87–96 (2007)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kutyniok, G., Labate, D.: Introduction to shearlets. In: Kutyniok, G., Labate, D. (eds.) Multiscale Analysis for Multivariate Data, pp. 1–38. Birkhäuser, Basel (2012)zbMATHGoogle Scholar
  15. 15.
    Labate, D., Weiss, G.: Continuous and discrete reproducing systems that arise from translations. Theory and applications of composite wavelets. In: Forster, B., Massopust, P. (eds.) Four Short Courses on Harmonic Analysis, pp. 87–130. Birkhäuser, Basel (2010)CrossRefGoogle Scholar
  16. 16.
    MacArthur, J.: Compatible dilations and wavelets for the wallpaper groups. Ph.D. thesis, Dalhousie University (in preparation)Google Scholar
  17. 17.
    MacArthur, J., Taylor, K.: Wavelets with crystal symmetry shifts. J. Fourier Anal. Appl. 17, 1109–1118 (2011)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Mackey, G.: The Theory of Unitary Group Representations. University of Chicago Press, Chicago (1976)zbMATHGoogle Scholar
  19. 19.
    Schattschneider, D.: The plane symmetry groups, their recognition and notation. Am. Math. Mon. 85, 439–450 (1978)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Taylor, K.: C-algebras of crystal groups. Oper. Theory: Adv. Appl. 41, 511–518 (1989)MathSciNetGoogle Scholar

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Authors and Affiliations

  • Kathy D. Merrill
    • 1
  1. 1.Department of MathematicsThe Colorado CollegeColorado SpringsUSA

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