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Mediterranean Journal of Mathematics

, Volume 10, Issue 1, pp 353–365 | Cite as

The Continuous Zak Transform and Generalized Gabor Frames

  • Ali Akbar Arefijamaal
Article

Abstract

Let G be a locally compact abelian group and H be a closed (not necessarily discrete) subgroup of G. In this article, we introduce the notion of Zak transform associated to H and obtain a necessary and sufficient condition to generate continuous Gabor frames for L 2(G). These results can be extended to non-abelian locally compact groups which are semidirect products. As an application, we obtain a characterization of admissible vectors for the regular and quasi regular representations.

Mathematics Subject Classification (2010)

Primary 43A32 Secondary 43A70 

Keywords

LCA groups Fourier transform Zak transform continuous frame regular representation semidirect product groups 

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References

  1. 1.
    Ali S.T., Antoine J.P., Gazeau J.P.: Coherent states, wavelets and their generalizations, Graduate Texts in Contemporary Physics. Springer-Verlag, New York (2000)CrossRefGoogle Scholar
  2. 2.
    Ali S.T., Antoine J.P., Gazeau J.P.: Continuous frames in Hilbert spaces. Ann. Physics 222, 1–37 (1993)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Arefijamaal A.A., Kamyabi-Gol R.A.: On the square integrability of quasi regular representation on semidirect product groups. J. Geom. Anal. 19(3), 541–552 (2009)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    A. A. Arefijamaal, R. A. Kamyabi-Gol, R. Raisi Tousi and N. Tavallaee, Anew approach to continuous Riesz bases, preprint.Google Scholar
  5. 5.
    Askari-Hemmat A., Dehghan M.A., Radjabalipour M.: Generalized frames and their redundancy. Proc. Amer. Math. Soc. 129(4), 1143–1147 (2001)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Casazza P.G., Kutyniok G., Li S.: Fusion frames and distributed processing. Appl. Comput. Harmon. Anal. 25(1), 114–132 (2008)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Folland G.B.: A course in abstract harmonic analysis. CRC Press, Boca Raton (1995)MATHGoogle Scholar
  8. 8.
    Führ H.: Admissible vectors for the regular representation. Proc. Amer. Math. Soc. 130(10), 2959–2970 (2002)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    H.Führ, Abstract harmonic analysis of continuous wavelet transforms, Springer Lecture Notes in Mathematics, no. 1863, Berlin, 2005.Google Scholar
  10. 10.
    Führ H., Mayer M.: Continuous wavelet transforms from semidirect products: cyclic representations and Plancherel measure. J. Fourier Anal. Appl. 8(4), 375–397 (2002)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Gabardo J.P., Han D.: Frames associated with measurable space. Adv. Comp. Math. 18(3), 127–147 (2003)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    A. Ghaani Farashahi, A new approach to the Fourier analysis on semi-direct products of groups, preprint (arXiv:math/1201.1179v1).Google Scholar
  13. 13.
    K. Gröchenig, Aspects of Gabor analysis on locally compact abelian groups, in: Gabor Analysis and Algorithms, 211–231, Appl. Numer. Harmon. Anal., Birkhäuser, Boston, MA, 1998.Google Scholar
  14. 14.
    E. Hewitt and K. A. Ross, Abstract harmonic analysis, Vol 1., Springer-Verlag, Berlin, 1970.Google Scholar
  15. 15.
    G. Hochschild, The structure of Lie groups, Holden-Day, Inc., San Francisco- London-Amsterdam, 1965.Google Scholar
  16. 16.
    Janssen A.J.E.M.: The Zak transform: a signal transform for sampled timecontinuous signals. Philips J. Res. 43(1), 23–69 (1988)MathSciNetMATHGoogle Scholar
  17. 17.
    Kutyniok G.: The Zak transform on certain locally compact groups. J. Math. Sci. (N.S.) (Delhi) 1, 62–85 (2002)MathSciNetMATHGoogle Scholar
  18. 18.
    Kutyniok G.: A qualitative uncertainty principle for functions generating a Gabor frame on LCA groups. J. Math. Anal. Appl. 279, 580–596 (2003)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Kaniuth E., Kutyniok G.: Zeros of the Zak transforms on locally compact abelian groups. Proc. Amer. Math. Soc. 126, 3561–3569 (1998)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Pevnyi A., Zheludev V.: Construction of wavelet analysis in the space of discrete splines using Zak transform. J. Fourier Anal. Appl. 8(1), 59–83 (2002)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    H. Reiter and J. D. Stegeman, Classical harmonic analysis and locally compact groups, London Math. Soc. Monogr. 22, Oxford Univ. Press, 2000.Google Scholar
  22. 22.
    Gh. Sadeghi, A. A. Arefijamaal, von Neumann-Schatten frames in separable Banach spaces, to appear in Mediterr. J. Math.Google Scholar
  23. 23.
    Sun W.: G-frames and G-Riesz bases. J. Math. Anal. Appl. 322, 437–452 (2006)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    E. Weber, Wavelet transforms and admissible group representations, in: Representations, Wavelets, and Frames, Appl. Numer. Harmon. Anal., Birkhäuser, Boston (2008), 47–67.Google Scholar
  25. 25.
    Weil A.: Sur certains groups d’operateurs unitaires. Acta Math. 111, 143–211 (1964)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    M. W. Wong, Wavelet transforms and localization operators, Operator Theory: Advances and Applications 136, Birkhäuser Verlag, Basel, 2002.Google Scholar
  27. 27.
    Zak J.: Finite translation in solid state physics. Phys. Rev. Letters 19, 1385–1387 (1967)CrossRefGoogle Scholar
  28. 28.
    Zhang S., Vourdas A.: Analytic representation of finite quantum systems. J. Phys. A 37, 8349–8363 (2004)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Department of Mathematics and Computer SciencesSabzevar Tarbiat Moallem UniversitySabzevarIran

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