Skip to main content
SpringerLink
Log in

Localization Operators Associated with the Hypergeometric Wigner Transform Related to the Cherednik Operators in the Case of the Root System \(BC_{d}\)

  • Original Paper
  • Published:
Bulletin of the Iranian Mathematical Society Aims and scope Submit manuscript

Abstract

Localization operators have a relatively recent development in pure and applied mathematics. Motivated by Wong’s approach, we will study in this paper the time–frequency analysis associated with the hypergeometric Wigner transform related to the Cherednik operators in the case of the root system \(BC_{d}\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, New York (1988)

    MATH  Google Scholar 

  2. Boggiatto, P., Wong, M.W.: Two-wavelet localization operators on \(L^{p}(\mathbb{R}^{d})\) for the Weyl-Heisenberg group. Integr. Equations Oper. Theory 49, 1–10 (2004)

    Article  Google Scholar 

  3. Calderon, J.P.: Intermediate spaces and interpolation, the complex method. Stud. Math. 24, 113–190 (1964)

    Article  MathSciNet  Google Scholar 

  4. Cherednik, I.: A unification of Knizhnik–Zamolod chnikove quations and Heckman–Opdam operators via affine Hecke algebras. Invent. Math. 106, 411–432 (1991)

    Article  MathSciNet  Google Scholar 

  5. Chettaoui, C., Hassini, A., Trimèche, K.: The hypergeometric Wigner and Weyl transforms attached to the Cherednik operators in the W-invariant case. Integr. Transform Spec. Funct. 28(9), 663–681 (2017)

    Article  MathSciNet  Google Scholar 

  6. Daubechies, I.: Time-frequency localization operators: a geometric phase space approach. IEEE Trans. Inf. Theory 34(4), 605–612 (1988)

    Article  MathSciNet  Google Scholar 

  7. Daubechies, I., Paul, T.: Time-frequency localization operators-a geometric phase space approach: II. The use of dilations. Inverse Probl. 4(3), 661–680 (1988)

    Article  Google Scholar 

  8. Daubechies, I.: The wavelet transform, time-frequency localization and signal analysis. IEEE Trans. Inf. Theory 36(5), 961–1005 (1990)

    Article  MathSciNet  Google Scholar 

  9. Folland, G.B.: Introduction to Partial Differential Equations, 2nd edn. Princeton University Press, Princeton (1995)

    MATH  Google Scholar 

  10. He, Z., Wong, M.W.: Localization operators associated to square integrable group representations. Panam. Math. J. 6(1), 93–104 (1996)

    MathSciNet  MATH  Google Scholar 

  11. Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge University Press, Cambridge (1990)

    Book  Google Scholar 

  12. Lieb, E.H.: Integral bounds for radar ambiguity functions and Wigner distributions. J. Math. Phys. 31(3), 594–599 (1990)

    Article  MathSciNet  Google Scholar 

  13. Liu, L.: A trace class operator inequality. J. Math. Anal. Appl. 328, 1484–1486 (2007)

    Article  MathSciNet  Google Scholar 

  14. Ma, B., Wong, M.W.: \(L^{p}-\)boundedness of wavelet multipliers. Hokkaido Math. J. 33, 637–645 (2004)

    Article  MathSciNet  Google Scholar 

  15. Mejjaoli, H., Trimèche, K.: Characterization of the support for the Hypergeometric Fourier transform of the \(W\)-invariant functions and distributions on \({\mathbb{R}}^{d}\) and Roe’s theorem. J. Inequal. Appl. (2014). https://doi.org/10.1186/1029-242X-2014-99

  16. Mejjaoli, H.: Qualitative uncertainty principles for the hypergeometric Fourier transform. Acta Math. Hung. 145(1), 229–251 (2015)

    Article  MathSciNet  Google Scholar 

  17. Olafsson, G., Pasquale, A.: Ramanujan’s master theorem for hypergeometric Fourier transform on root systems. J. Fourier Anal. Appl. 19(6), 1150–1183 (2013)

    Article  MathSciNet  Google Scholar 

  18. Opdam, E.M.: Harmonic analysis for certain representations of graded Hecke algebras. Acta. Math. 175, 75–121 (1995)

    Article  MathSciNet  Google Scholar 

  19. Opdam, E.M.: Lecture Notes on Dunkl operators for real and complex reflection groups. With a preface by Toshio Oshima, MSJ Memoirs, vol. 8. Mathematical Society of Japan, Tokyo (2000)

  20. Rösler, M.: Positive convolution structure for a class of Heckman-Opdam hypergeometric functions of type BC. J. Funct. Anal. 258, 2779–2800 (2010)

    Article  MathSciNet  Google Scholar 

  21. Schapira, B.: Contributions to the hypergeometric function theory of Heckman and Opdam:sharpe stimates, Schwartz spaces, heat kernel. Geom. Funct. Anal. 18(1), 222–250 (2008)

    Article  MathSciNet  Google Scholar 

  22. Stein, E.M.: Interpolation of linear operators. Trans. Am. Math. Soc. 83, 482–492 (1956)

    Article  MathSciNet  Google Scholar 

  23. Trimèche, K.: The Harmonic analysis associated to the Heckman-Opdam’s theory and its applications to a root system of a type \(BC_{d}\). Korean J. Math. 27(1), 221–267 (2019)

    MathSciNet  MATH  Google Scholar 

  24. Wong, M.W.: Localization operators on the Weyl-Heisenberg group. In: Pathak, R.S. (ed.) Geometry, Analysis and Applications, pp. 303–314. World-Scientific, Singapore (2001)

    Google Scholar 

  25. Wong, M.W.: \(L^{p}\) boundedness of localization operators associated to left regular representations. Proc. Am. Math. Soc. 130, 2911–2919 (2002)

    Article  Google Scholar 

  26. Wong, M.W.: Wavelet Transforms and Localization Operators, vol. 136. Springer Science & Business Media, New York (2002)

    Book  Google Scholar 

Download references

Acknowledgements

The authors are deeply indebted to the referees for providing constructive comments and helps in improving the contents of this article. The second author thanks the professor M.W. Wong for his help.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Tunis El Manar Faculty of Sciences of Tunis, Campus, 2092, Tunis, Tunisia

    Amina Hassini & Khalifa Trimèche

  2. Department of Mathematics, College of Sciences, Taibah University, PO BOX 30002, Al Madinah AL Munawarah, Saudi Arabia

    Hatem Mejjaoli

Authors
  1. Amina Hassini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hatem Mejjaoli
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Khalifa Trimèche
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hatem Mejjaoli.

Additional information

Communicated by Hamid Reza Ebrahimi Vishki.

Dedicated to the spirit of Ali Hassini.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hassini, A., Mejjaoli, H. & Trimèche, K. Localization Operators Associated with the Hypergeometric Wigner Transform Related to the Cherednik Operators in the Case of the Root System \(BC_{d}\). Bull. Iran. Math. Soc. 46, 1501–1531 (2020). https://doi.org/10.1007/s41980-019-00339-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41980-019-00339-8

Keywords

Mathematics Subject Classification

Search

Navigation