Advertisement

The Journal of Geometric Analysis

, Volume 12, Issue 1, pp 89–102 | Cite as

A characterization of the higher dimensional groups associated with continuous wavelets

  • R. S. LaugesenEmail author
  • N. Weaver
  • G. L. Weiss
  • E. N. Wilson
Article

Abstract

A subgroup D of GL (n, ℝ) is said to be admissible if the semidirect product of D and ℝ n , considered as a subgroup of the affine group on ℝ n , admits wavelets ψ ∈ L2(ℝ n ) satisfying a generalization of the Calderón reproducing, formula. This article provides a nearly complete characterization of the admissible subgroups D. More precisely, if D is admissible, then the stability subgroup Dx for the transpose action of D on ℝ n must be compact for a. e. x. ∈ ℝ n ; moreover, if Δ is the modular function of D, there must exist an a ∈ D such that |det a| ≠ Δ(a). Conversely, if the last condition holds and for a. e. x ∈ ℝ n there exists an ε > 0 for which the ε-stabilizer D x ε is compact, then D is admissible. Numerous examples are given of both admissible and non-admissible groups.

Math Subject Classification

42MCS2000 

Key Words and Phrases

continuous wavelets discrete wavelets admissable dialatin groups 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Ali, S. T., Antoine, J.-P., and Gazeau, J.-P.,Coherent States, Wavelets and Their Generalizations, Springer-Verlag, Inc., New York, (2000).zbMATHGoogle Scholar
  2. [2]
    Bacry, H., Grossman, A., and Zak, J.Geometry of Generalized Coherent States, Group Theoretical Methods in Physics (Fourth International Colloquium, Nijmegen, 1975), Springer-Verlag, Berlin, Lecture Notes in Physics,50, 249–268, (1978).Google Scholar
  3. [3]
    Bernier, D. and Taylor, K.F. Wavelets from square-integrable representations,SIAM J. Math. Anal.,27(2), 594–608, (1996).zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Bruna, J. Sampling in complex and harmonic analysis,Proc. of 3 rd Eur. Conf. in Mathematics, July, (2000).Google Scholar
  5. [5]
    Calderón, A.P. Intermediate spaces and interpolation, the complex method,Stud. Math.,24, 113–190, (1964).zbMATHGoogle Scholar
  6. [6]
    Carey, A.L. Square integrable representations of non-unimodular groups,Bull. Austral. Math. Soc.,15, 1–12, (1976).zbMATHMathSciNetCrossRefGoogle Scholar
  7. [7]
    Chui, C., Czaja, W., Maggioni, M., and Weiss, G.L. Characterization of general tight wavelet frames with matrix dilation and tightness preserving oversampling, to appear inJ. Four. Anal. and Applications.Google Scholar
  8. [8]
    Corbett, J. Coherent States on Kinematic Groups: The Study of Spatio-Temporal Wavelets and Their Applications to Motion Estimation, Ph.D. Thesis, Washington University, (1999).Google Scholar
  9. [9]
    Duflo, M. and Moore, C.C. On the representation of a nonunimodular locally compact group.J. Funct. Anal.,21, 209–243, (1976).zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    DeMari, F. and Nowak, K. Reproducing formulas and commutative operator algebras related to affine transformations of the time-frequency plane, preprint.Google Scholar
  11. [11]
    Führ, H., Wavelet frames and admissibility in higher dimensions,J. of Math. Phys.,37, 6353–6366, (1996).zbMATHCrossRefGoogle Scholar
  12. [12]
    Führ, H. Continuous wavelet transforms from semidirect products, to appear inRevista Ciencias Matematicas.Google Scholar
  13. [13]
    Führ, H. Admissible vectors for the regular representations, to appear inProc. Am. Math. Soc. Google Scholar
  14. [14]
    Führ, H. and Mayer, M. Continuous wavelet transforms from cyclic representations: A general approach using plancherel measure, preprint, (1999).Google Scholar
  15. [15]
    Heil, C.E. and Walnut, D.F. Continuous and discrete wavelet transforms,SIAM Review,31, 628–666, (1989).zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    Isham, C.J. and Klauder, J.R. Coherent states forn-dimensional Euclidean groups and their application,J. of Math. Phys.,32, 607–620, (1991).zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    Kalisa, C. Etats Cohérents Affines: Canoniques, Galiléns et Relativistes, Ph.D. Thesis, Université Louvaine, (1993).Google Scholar
  18. [18]
    Liu, H. and Peng, L. Admissible wavelets associated with the Heisenberg group,Pacific J. of Math.,180, 101–123, (1997).MathSciNetCrossRefGoogle Scholar
  19. [19]
    Weiss, G.L. and Wilson, E.N.The Mathematical Theory of Wavelets, Proceedings of the NATO-ASI meeting, Harmonic Analysis 2000-A Celebration, Kluwer Publisher, to appear in 2001.Google Scholar
  20. [20]
    Zimmer, R.J.Ergodic Theory and Semisimple Groups, Birkhäuser, Boston, (1980).Google Scholar

Copyright information

© Mathematica Josephina, Inc. 2002

Authors and Affiliations

  • R. S. Laugesen
    • 1
    Email author
  • N. Weaver
    • 2
  • G. L. Weiss
    • 2
  • E. N. Wilson
    • 2
  1. 1.Department of MathematicsUniversity of IllinoisUrbana
  2. 2.Department of MathematicsWashington UniversitySt. Louis

Personalised recommendations