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Weyl transform and Weyl multipliers associated with locally compact abelian groups

  • R. Radha
  • N. Shravan Kumar
Article
  • 131 Downloads

Abstract

In this paper, we extend to the context of locally compact abelian groups, the notion of Weyl transform. We also define and characterize Weyl multipliers for certain function spaces. A dual space characterization is provided for the space of Weyl multipliers on \(L^p\) spaces. Finally we also study the twisted shift-invariant subspaces.

Keywords

Locally compact abelian group Weyl transform Twisted convolution Twisted Figà-Talamanca Herz algebra Twisted shift-invariant spaces 

Mathematics Subject Classification

Primary 43A15 43A25 Secondary 22B05 43A32 

Notes

Acknowledgements

The authors would like to thank the unknown referee for meticulously reading the paper and giving us some valuable suggestions. The proof of Proposition 2.8 (c) was suggested by Prof. Rajaram Bhat through a personal communication. We thank Prof. Rajaram Bhat (Indian Statistical Institute, Bangalore) for his kind response to our query.

References

  1. 1.
    Aldroubi, A., Gröchenig, K.: Nonuniform sampling and reconstruction in shift-invariant spaces. SIAM Rev. 43, 584–620 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bownik, M.: The structure of shift-invariant subspaces of \(L^2(\mathbb{R}^n)\). J. Funct. Anal. 177, 282–309 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cabrelli, C., Paternostro, V.: Shift-invariant spaces on LCA groups. J. Funct. Anal. 258, 2034–2059 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cowling, M.: The predual of the space of convolutors on a locally compact group. Bull. Aust. Math. Soc. 57, 409–414 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Davidson, K.R.: Nest Algebras, Pitman Research Notes in Mathematics, Vol. 191. Longman Scientific and Technical, Burnt Mill Harlow, Essex, UK (1988)Google Scholar
  6. 6.
    Derighetti, A.: Convolution operators on groups. Lecture Notes of the Unione Matematica Italiana, vol. 11. Springer (2011)Google Scholar
  7. 7.
    Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  8. 8.
    Feldman, J., Greenleaf, F.P.: Existence of Borel transversals in groups. Pac. J. Math. 25, 455–461 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Figà-Talamanca, A.: Translation invariant operators in \(L^p,\). Duke Math. J. 32, 495–501 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Figà-Talamanca, A., Gaudry, G.I.: Density and representation theorems for multipliers of type \((p, q),\). J. Aust. Math. Soc. 7, 1–6 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Folland, G.B.: Harmonic Analysis in Phase Space. Princeton University Press, Princeton (1989)zbMATHGoogle Scholar
  12. 12.
    Folland, G.B.: A Course in Abstract Harmonic Analysis. CRC Press, Boca Raton (1995)zbMATHGoogle Scholar
  13. 13.
    Gaudry, G.I.: Quasimeasures and operators commuting with convolution. Pac. J. Math. 18, 461–476 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Herz, C.: Harmonic synthesis for subgroups. Ann. Inst. Fourier (Grenoble) 23, 91–123 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Katznelson, Y.: An Introduction to Harmonic Analysis. Cambridge University Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar
  16. 16.
    Kutyniok, G.: The Zak transform on certain locally compact groups. J. Math. Sci. 1, 62–85 (2002)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Kutyniok, G.: Linear independence of time-frequency shifts under a generalized Schrödinger representation. Arch. Math. 78, 135–144 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Larsen, R.: An Introduction to the Theory of Multipliers. Springer, New York (1971)CrossRefzbMATHGoogle Scholar
  19. 19.
    Mauceri, G.: The Weyl transform and bounded operators on \(L^p(\mathbb{R}^n),\). J. Funct. Anal. 39, 408–429 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Radha, R., Thangavelu, S.: Weyl multipliers for invariant Sobolev spaces. Proc. Indian Acad. Sci. (Math. Sci.) 108, 31–40 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Raghunathan, M.S.: Discrete Subgroups of Lie Groups. Springer, Berlin (1972)CrossRefzbMATHGoogle Scholar
  22. 22.
    Reiter, H., Stegeman, J.D.: Classical Harmonic Analysis and Locally Compact Groups. Clarendon Press, Oxford (2007)zbMATHGoogle Scholar
  23. 23.
    Rieffel, M.A.: Multipliers and tensor products of \(L^p\)-spaces of locally compact groups. Studia Math. 33, 71–82 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Thangavelu, S.: Harmonic analysis on the Heisenberg group. Progress in Mathematics, vol 159. Birkhauser, Boston (1998)Google Scholar
  25. 25.
    Weil, A.: Sur certains groupes d’opérateurs unitaires. Acta Math. 111, 143–211 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Wendel, J.G.: Left centralizers and isomorphisms of group algebras. Pac. J. math. 2, 251–261 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Wong, M.W.: Weyl Transforms. Springer, New York (1998)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology MadrasChennaiIndia
  2. 2.Department of MathematicsIndian Institute of Technology DelhiNew DelhiIndia

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