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Indian Journal of Pure and Applied Mathematics

, Volume 50, Issue 4, pp 1115–1132 | Cite as

Standard shearlet group in arbitrary space dimensions and projections in its L1-algebra

  • Masoumeh ZareEmail author
  • Rajab Ali Kamyabi-GolEmail author
Article
  • 10 Downloads

Abstract

This paper is devoted to definition standard shearlet group \(\mathbb{S} = {\mathbb{R}^ + } \times {\mathbb{R}^{n - 1}} \times {\mathbb{R}^n}\), in arbitrary space dimensions and concerned with the projections which are, self adjoint idempotents in the group algebra \({L^1}(\mathbb{S})\). Actually we determine minimal projections, associated with an open free orbit, in details.

Key words

L1-projection shearlet group square-integrable representation admissible function 

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References

  1. 1.
    S. T. Ali, J. P. Antoine, and J. P. Gazeau, Coherent states, wavelets and their generalizations, New York: Springer-Verlag, (2000).CrossRefGoogle Scholar
  2. 2.
    S. Dahlke and G. Teschke, The continuous shearlet transform in arbitrary space dimensions, J. Fourier Anal. Appl., 16 (2010), 340–364.MathSciNetCrossRefGoogle Scholar
  3. 3.
    S. Dahlke, G. Kutyniok, G. Steidl, and G. Teschke, Shearlet coorbit spaces and associated Banach frames, Appl. Comput. Harm. Anal., 27(2) (2009), 195–214.MathSciNetCrossRefGoogle Scholar
  4. 4.
    J. Dixmier, C*-algebras, North-Holland Publishing Co., Amsterdam-New York-Oxford, (1977), Translated from the French by Francis Jellett, North-Holland Mathematical Library, Vol. 15.Google Scholar
  5. 5.
    G. B. Folland, A course in abstract harmonic analysis, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, (1995).zbMATHGoogle Scholar
  6. 6.
    P. Eymard and M. Terp, La transformation de Fourier et son inverse sur le groupe de ax + b d’un corps local, Analyse harmonique sur les groups de Lie, Lecture Notes in Math., 739, Springer, Berlin, (1979), 207–248.CrossRefGoogle Scholar
  7. 7.
    K. Grochenig, E. Kaniuth, and K. F. Taylor, Compact open sets in duals and projections in L1-algebras of certain semi-direct product groups, Math. Proc. Cambridge Philos. Soc., Ill (1992), 545–556.CrossRefGoogle Scholar
  8. 8.
    K. Guo, G. Kutyniok, and D. Labate, Sparse multidimensional representations using anisotropic dilation and shear operators, Wavelets and splines: Athens 2005, pp. 189–201, Mod. Methods Math., Nashboro Press, Brentwood, TN, (2006).Google Scholar
  9. 9.
    H. Helson, Note on harmonic functions, Proc. Amer. Math. Soc., 4 (1953), 686–691.MathSciNetCrossRefGoogle Scholar
  10. 10.
    E. Hewitt and K. A. Ross, Abstract harmonic analysis, Vol. 1, Springer-Verlag, Berlin, (1963).Google Scholar
  11. 11.
    E. Kaniuth and Keith F. Taylor, Induced representations of locally compact groups, volume 197 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, (2013).Google Scholar
  12. 12.
    E. Kaniuth and K. F. Taylor, Projections in C*-algebras of nilpotent groups, Manuscripta Math., 65 (1989), 93–111.MathSciNetCrossRefGoogle Scholar
  13. 13.
    E. Kaniuth and K. F. Taylor, Minimal projections in L1-algebras and open points in the dual spaces of semi-direct product groups, J. London Math. Soc. (2), 53(1) (1996), 141–157.MathSciNetCrossRefGoogle Scholar
  14. 14.
    G. W. Mackey, Borel structures in groups and their duals, Trans. Amer. Math. Soc., (1957), 134–165.Google Scholar
  15. 15.
    W. Rudin, Idempotent measures on Abelian groups, Pacific J. Math., 9 (1959), 195–209.MathSciNetCrossRefGoogle Scholar

Copyright information

© Indian National Science Academy 2019

Authors and Affiliations

  1. 1.Department of Pure MathematicsFerdowsi University of Mashhad and Center of Excellence in Analysis on Algebraic Structures (CEAAS)MashhadIran

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