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Weyl transform and Weyl multipliers associated with locally compact abelian groups

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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.

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References

  1. Aldroubi, A., Gröchenig, K.: Nonuniform sampling and reconstruction in shift-invariant spaces. SIAM Rev. 43, 584–620 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bownik, M.: The structure of shift-invariant subspaces of \(L^2(\mathbb{R}^n)\). J. Funct. Anal. 177, 282–309 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cabrelli, C., Paternostro, V.: Shift-invariant spaces on LCA groups. J. Funct. Anal. 258, 2034–2059 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cowling, M.: The predual of the space of convolutors on a locally compact group. Bull. Aust. Math. Soc. 57, 409–414 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Davidson, K.R.: Nest Algebras, Pitman Research Notes in Mathematics, Vol. 191. Longman Scientific and Technical, Burnt Mill Harlow, Essex, UK (1988)

  6. Derighetti, A.: Convolution operators on groups. Lecture Notes of the Unione Matematica Italiana, vol. 11. Springer (2011)

  7. Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators. Cambridge University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  8. Feldman, J., Greenleaf, F.P.: Existence of Borel transversals in groups. Pac. J. Math. 25, 455–461 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  9. Figà-Talamanca, A.: Translation invariant operators in \(L^p,\). Duke Math. J. 32, 495–501 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  11. Folland, G.B.: Harmonic Analysis in Phase Space. Princeton University Press, Princeton (1989)

    MATH  Google Scholar 

  12. Folland, G.B.: A Course in Abstract Harmonic Analysis. CRC Press, Boca Raton (1995)

    MATH  Google Scholar 

  13. Gaudry, G.I.: Quasimeasures and operators commuting with convolution. Pac. J. Math. 18, 461–476 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  14. Herz, C.: Harmonic synthesis for subgroups. Ann. Inst. Fourier (Grenoble) 23, 91–123 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  15. Katznelson, Y.: An Introduction to Harmonic Analysis. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  16. Kutyniok, G.: The Zak transform on certain locally compact groups. J. Math. Sci. 1, 62–85 (2002)

    MathSciNet  MATH  Google Scholar 

  17. Kutyniok, G.: Linear independence of time-frequency shifts under a generalized Schrödinger representation. Arch. Math. 78, 135–144 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Larsen, R.: An Introduction to the Theory of Multipliers. Springer, New York (1971)

    Book  MATH  Google Scholar 

  19. Mauceri, G.: The Weyl transform and bounded operators on \(L^p(\mathbb{R}^n),\). J. Funct. Anal. 39, 408–429 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  20. Radha, R., Thangavelu, S.: Weyl multipliers for invariant Sobolev spaces. Proc. Indian Acad. Sci. (Math. Sci.) 108, 31–40 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Raghunathan, M.S.: Discrete Subgroups of Lie Groups. Springer, Berlin (1972)

    Book  MATH  Google Scholar 

  22. Reiter, H., Stegeman, J.D.: Classical Harmonic Analysis and Locally Compact Groups. Clarendon Press, Oxford (2007)

    MATH  Google Scholar 

  23. Rieffel, M.A.: Multipliers and tensor products of \(L^p\)-spaces of locally compact groups. Studia Math. 33, 71–82 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  24. Thangavelu, S.: Harmonic analysis on the Heisenberg group. Progress in Mathematics, vol 159. Birkhauser, Boston (1998)

  25. Weil, A.: Sur certains groupes d’opérateurs unitaires. Acta Math. 111, 143–211 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  26. Wendel, J.G.: Left centralizers and isomorphisms of group algebras. Pac. J. math. 2, 251–261 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  27. Wong, M.W.: Weyl Transforms. Springer, New York (1998)

    MATH  Google Scholar 

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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.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Madras, Chennai, 600036, India

    R. Radha

  2. Department of Mathematics, Indian Institute of Technology Delhi, New Delhi, Delhi, 110016, India

    N. Shravan Kumar

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  1. R. Radha
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  2. N. Shravan Kumar
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Correspondence to R. Radha.

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Radha, R., Shravan Kumar, N. Weyl transform and Weyl multipliers associated with locally compact abelian groups. J. Pseudo-Differ. Oper. Appl. 9, 229–245 (2018). https://doi.org/10.1007/s11868-017-0213-0

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  • DOI: https://doi.org/10.1007/s11868-017-0213-0

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