Abstract
The wavelet group and wavelet representation associated with shifts coming from a two dimensional crystal symmetry group \(\Gamma \) and dilations by powers of 3, are defined and studied. The main result is an explicit decomposition of this \(3\Gamma \)-wavelet representation into irreducible representations of the wavelet group. Because we prove that the \(3\Gamma \)-wavelet representation is multiplicity free, this direct integral decomposition is essentially unique.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Auslander, L.: An account of the theory of crystallographic groups. Proc. Am. Math. Soc. 16, 1230–1236 (1965)
Currey, B., Mayeli, A., Oussa, V.: Decompositions of generalized wavelet representations. Contemp. Math. 626, 67–85 (2014)
Dai, X., Larson, D.R.: Wandering vectors for unitary systems and orthogonal wavelets. Mem. Am. Math. Soc. 134 (640) (1998)
Dai, X., Larson, D.R., Speegle, D.: Wavelet sets in \(\mathbb{R}^n\). J. Fourier Anal. Appl. 3, 451–456 (1997)
Dutkay, D.E.: Low-pass filters and representations of the Baumslag Solitar group. Trans. Am. Math. Soc. 358, 5271–5291 (2006)
Dutkay, D.E., Han, D., Jorgensen, P.E.T., Picioroaga, G.: On common fundamental domains. Adv. Math. 239, 109–127 (2013)
Dutkay, D.E., Silvestrov, S.: Reducibility of the wavelet representation associated to the Cantor set. Proc. Am. Math. Soc. 139, 3657–3664 (2011)
Federov, E.: Symmetry in the plane. Zapiski Rus. Mineralog. Obščestva, Ser. 2 28 (1891), 345–390
Folland, G.: A Course in Abstract Harmonic Analysis. CRC Press, Boca Raton (1995)
González, A.L., del Carmen Moure, M.: Crystallographic Haar wavelets. J. Fourier Anal. Appl. 17, 1119–1137 (2011)
Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis. Springer, New York (1979)
Hulanicki, A.: Means and Folner conditions on locally compact groups. Studia Math. 27, 87–104 (1966)
Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras, Vol. II, Advanced Theory. Pure and Applied Mathematics, vol. 100. Academic Press, Inc., Orlando, FL (1986)
Kaniuth, E.: Der Typ der regulären Darstellung diskreter Gruppen. Math. Ann. 182, 334–339 (1969)
Kaniuth, E., Taylor, K.F.: Induced Representations of Locally Compact Groups. Cambridge Tracts in Mathematics v., vol. 197. Cambridge University Press, Cambridge (2012)
Lim, L.-H., Packer, J.A., Taylor, K.F.: A direct integral decomposition of the wavelet representation. Proc. Am. Math. Soc. 129, 3057–3067 (2001)
MacArthur, J., Taylor, K.F.: Wavelets with crystal symmetry shifts. J. Fourier Anal. Appl. 17, 1109–1118 (2011)
Mackey, G.W.: Imprimitivity for representations of locally compact groups I. Proc. Natl. Acad. Sci. USA 35, 537–545 (1949)
Mackey, G.W.: The Theory of Unitary Group Representations. Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1976)
Merrill, K.D.: Simple Wavelet Sets for Scalar Dilations in \(\mathbb{R}^2\), Representations, Wavelets, and Frames. Applied and Numerical Harmonic Analysis, pp. 177–192. Birkhäuser, Boston (2008)
Merrill, K.D.: Simple wavelet sets for matrix dilations in \(\mathbb{R}^2\). Numer. Funct. Anal. Optim. 33, 1112–1125 (2012)
Merrill, K.D.: Simple wavelet sets in \(\mathbb{R}^n\). J. Geom. Anal. 25, 1295–1305 (2015)
Merrill, K.D.: Wavelet Sets for Crystallographic Groups. To appear in Excursions in Harmonic Analysis, vol. 6. Birkhäuser, Basel (2021)
Morandi, P.J.: The Classification of Wallpaper Patterns: From Cohomology to Escher’s Tesselations. New Mexico State University, Las Cruces (2007)
Nielsen, O.A.: Direct Integral Theory. Lecture Notes in Pure and Applied Mathematics, vol. 61. Marcel Dekker Inc, New York (1980)
Paterson, A.L.T.: Amenability, Mathematical Surveys and Monographs, vol. 29. American Mathematical Society, Providence, RI (1988)
Schattschneider, D.: The plane symmetry groups, their recognition and notation. Am. Math. Mon. 85, 439–450 (1978)
Acknowledgements
The third author was supported by an individual grant from the Simons Foundation (# 316981). We thank the anonymous referees for helpful suggestions and remarks.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Hartmut Führ.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Baggett, L.W., Merrill, K.D., Packer, J.A. et al. Decomposing the Wavelet Representation for Shifts by Wallpaper Groups. J Fourier Anal Appl 27, 16 (2021). https://doi.org/10.1007/s00041-021-09818-1
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00041-021-09818-1