Advertisement

The Journal of Geometric Analysis

, Volume 25, Issue 2, pp 1045–1074 | Cite as

Characterizing Abelian Admissible Groups

  • J. Bruna
  • J. Cufí
  • H. Führ
  • M. MiróEmail author
Article

Abstract

By definition, admissible matrix groups are those that give rise to a wavelet-type inversion formula. This paper investigates necessary and sufficient admissibility conditions for abelian matrix groups. We start out by deriving a block diagonalization result for commuting real-valued matrices. We then reduce the question of deciding admissibility to the subclass of connected and simply connected groups and derive a general admissibility criterion for exponential solvable matrix groups. For abelian matrix groups with real spectra, this yields an easily checked necessary and sufficient characterization of admissibility. As an application, we sketch a procedure for checking admissibility of a matrix group generated by finitely many commuting matrices with positive spectra.

We also present examples showing that the simple answers that are available for the real spectrum case fail in the general case.

An interesting byproduct of our considerations is a method that allows for an abelian Lie subalgebra \(\mathfrak{h} \subset gl(n,\mathbb{R})\) to check whether \(H = \exp(\mathfrak{h})\) is closed.

Keywords

Abelian admissible groups Quasiregular representation Wavelet transforms Calderón condition Fundamental region Lie algebra Lie group 

Mathematics Subject Classification (2000)

42C40 

Notes

Acknowledgements

HF would like to thank the Universitat Autónoma de Barcelona, Departament de Matemàtiques, for its hospitality.

References

  1. 1.
    Aniello, P., Cassinelli, G., De Vito, E., Levrero, A.: On discrete frames associated with semidirect products. J. Fourier Anal. Appl. 7, 199–206 (2001) CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bernat, P., Conze, N., Duflo, M., Lévy-Nahas, M., Raïs, M., Renouard, P., Vergne, M.: Représentations des Groupes de Lie Résolubles, Monographies de la Société Mathématique de France. Dunod, Paris (1972) zbMATHGoogle Scholar
  3. 3.
    Bernier, D., Taylor, K.: Wavelets from square-integrable representations. SIAM J. Math. Anal. 27, 594–608 (1996) CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bohnke, G.: Treillis d’ondelettes aux groupes de Lorentz. Ann. Inst. Henri Poincaré 54, 245–259 (1991) zbMATHMathSciNetGoogle Scholar
  5. 5.
    Corwin, L.J., Greenleaf, F.P.: Representations of Nilpotent Lie Groups and Their Applications. Part I. Basic Theory and Examples. Cambridge Studies in Advanced Mathematics, vol. 18. Cambridge University Press, Cambridge (1990) Google Scholar
  6. 6.
    Effros, E.G.: Transformation groups and C -algebras. Ann. Math. (2) 81, 38–55 (1965) CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Führ, H.: Wavelet frames and admissibility in higher dimensions. J. Math. Phys. 37, 6353–6366 (1996) CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Führ, H.: Continuous wavelet transforms with Abelian dilation groups. J. Math. Phys. 39, 3974–3986 (1998) CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Führ, H., Mayer, M.: Continuous wavelet transforms from semidirect products: cyclic representations and Plancherel measure. J. Fourier Anal. Appl. 8, 375–398 (2002) CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Führ, H.: Abstract Harmonic Analysis of Continuous Wavelet Transforms. Lecture Notes in Mathematics, vol. 1863. Springer, Heidelberg (2005) zbMATHGoogle Scholar
  11. 11.
    Führ, H.: Generalized Calderón conditions and regular orbit spaces. Colloq. Math. 120, 103–126 (2010) CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis I. Springer, Berlin (1963) CrossRefzbMATHGoogle Scholar
  13. 13.
    Neeb, K.-H., Hilgert, J.: Lie-Gruppen und Lie-Algebren. Vieweg, Wiesbaden (1991) zbMATHGoogle Scholar
  14. 14.
    Larson, D., Schulz, E., Speegle, D., Taylor, K.F.: Explicit cross-sections of singly generated group actions. In: Harmonic Analysis and Applications, Appl. Numer. Harmon. Anal., pp. 209–230. Birkhäuser, Boston (2006) CrossRefGoogle Scholar
  15. 15.
    Laugesen, R.S., Weaver, N., Weiss, G., Wilson, E.N.: Continuous wavelets associated with a general class of admissible groups and their characterization. J. Geom. Anal. 12, 89–102 (2002) CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Miró, M.: Funcions admissibles i ondetes ortonormals a \(\mathbb{R}^{n}\). Thesis, Universitat Autònoma de Barcelona (2010) Google Scholar
  17. 17.
    Murenzi, R.: Ondelettes multidimensionelles et application à l’analyse d’images. Thèse, Université Catholique de Louvain, Louvain-La-Neuve (1990) Google Scholar
  18. 18.
    Suprunenko, D.A., Tyshkevich, R.I.: Commutative Matrices. Academic Press, New York (1968) Google Scholar
  19. 19.
    Varadarajan, V.S.: Lie Groups, Lie Algebras, and Their Representations. Prentice-Hall, Englewood Cliffs (1974). 22-01 (17-02 22EXX) zbMATHGoogle Scholar
  20. 20.
    Wüstner, M.: On closed Abelian subgroups of real Lie groups. J. Lie Theory 7, 279–285 (1997) zbMATHMathSciNetGoogle Scholar
  21. 21.
    Zimmer, R.J.: Ergodic Theory and Semisimple Groups. Birkhäuser, Boston (1984) CrossRefzbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2013

Authors and Affiliations

  1. 1.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterra-BarcelonaSpain
  2. 2.Lehrstuhl A für MathematikRWTH AachenAachenGermany

Personalised recommendations