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Weyl Transforms Associated with Dunkl Wavelet Transform

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Abstract

In this paper, we define the Weyl transform associated with Dunkl wavelet transform and discuss its boundedness and compactness on the Lebesgue spaces.

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References

  1. Abdelkefi C., Amri B., Sifi, M.: Pseudo-differential operator associated with the Dunkl operator, Differential Integral Equations. 20(9) , 1035–1051 (2007)

  2. Al Zahrani, E.A., Mourou, M.A.: The continuous wavelet transform associated with a Dunkl type operator on the real line. Adv. Pure Math. 3(5), 443–450 (2013)

    Article  Google Scholar 

  3. Amri, B., Gasmi, A., Sifi, M.: Linear and Bilinear Multiplier Operators for the Dunkl Transform. Mediterr. J. Math. 7, 503–521 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Amri, B., Mustapha, S., Sifi, M.: On the boundedness of pseudo-differential operators associated with the Dunkl transform on the real line. Adv. Pure Appl. Math. 2(1), 89–107 (2010)

    MathSciNet  MATH  Google Scholar 

  5. Andersen, N.B.: Real Paley-Wiener theorems for the Dunkl transform on \(\mathbb{R} \). Integral Transf. Spec. Funct. 17(8), 543–547 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Daubechies I.: Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, Philadelphia, (1992)

  7. Dunkl, C.F.: Differential-difference operators associated to reflection groups. Trans. Amer. Math. Soc. 311, 167–183 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dunkl, C.F.: Hankel transform associated to finite reflection groups. Contemp. Math. 138, 123–138 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dunkl, C.F.: Integral kernel with reflection group invariance. Canad. J. Math. 43, 1213–1227 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dziri, M., Sahbani, S.: An inversion theorem for Dunkl transform. Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. 37(4), 33–41 (2017)

    MathSciNet  MATH  Google Scholar 

  11. Grossmann, A., Morlet, J.: Decomposition of Hardy functions intosquare integrable wavelets of constant shape. SIAM J. Math. Anal. 15(4), 723–736 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  12. Guliyev, V.S., Mammadov, Y.Y.: \((L_p, L_q)\) Boundedness of the fractional maximal operator associated with the Dunkl operator on the real line. Integral Transf. Spec. Funct. 21(8), 629–639 (2010)

    Article  MATH  Google Scholar 

  13. Li, Z., Liao, J.: Harmonic analysis associated with the one-dimensional Dunkl transform. Constr. Approx. 37(2), 233–281 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ma, R., Peng, L.: Weyl transforms of wavelets. Progr. Nat. Sci. 13(7), 489–496 (2003)

    MathSciNet  Google Scholar 

  15. Mejjaoli, H., Sraieb, N.: Uncertainty principles for the continuous Dunkl Gabor transform and the Dunkl continuous wavelet transform. Mediterr. J. Math. 5, 443–466 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Mejjaoli, H., Sraieb, N.: New uncertainty principles for the Dunkl wavelet transform. Int. J. Open Prob. Complex Anal. 12, 51–75 (2020)

    MATH  Google Scholar 

  17. Meyer, Y.: Wavelets and Operators. Cambridge University Press, Cambridge (1992)

    MATH  Google Scholar 

  18. Mourou, M.A.: Inversion of the dual Dunkl-Sonine transform on \(\mathbb{R} \) using Dunkl wavelets. SIGMA 5(71), 12 (2009)

    MathSciNet  MATH  Google Scholar 

  19. Peng, L., Ma, R.: Wavelets associated with Hankel transform and their Weyl transforms. Sci. China Ser. A-Math. 47, 393–400 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Pool, J.C.T.: Mathematical aspect of the Weyl correspondence. J. Math. Phys. 7, 66–76 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  21. Rachdi, L.T., Trimèche, K.: Weyl transforms associated with the spherical mean operator. Anal. Appl. 1(2), 141–164 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Simon, B.: The Weyl transforms and \(L^p\) functions on phase space. Proc. Amer. Math. Soc. 116, 1045–1047 (1992)

    MathSciNet  MATH  Google Scholar 

  23. Soltani, F.: \(L^p\)-Fourier multipliers for the Dunkl operator on the real line. J. Funct. Anal. 209, 16–35 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Soltani, F.: Inversion formula for the Dunkl-Wigner transform and compactness property for the Dunkl-Weyl transforms. J. Math. Res. Appl. 35(4), 425–434 (2015)

    MathSciNet  MATH  Google Scholar 

  25. Stein, E.M.: Interpolation of linear operators. Trans. Amer. Math. Soc. 83(2), 482–492 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  26. Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, New Jersey (1971)

    MATH  Google Scholar 

  27. Verma, S.K., Prasad, A.: Characterization of Weyl operator in terms of Mehler-Fock transform. Math. Methods Appl. Sci. 43, 9110–9128 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  28. Weyl, H.: The Theory of Groups and Quantum Mechanics. Dover, New York (1950)

    MATH  Google Scholar 

  29. Wong, M.W.: Weyl Transform. Springer-Verlag, New York (1998)

    Google Scholar 

  30. Zhao, J., Peng, L.: Wavelet and Weyl transform associated with the spherical mean operator. Integral Equations Operator Theory. 50(2), 279–290 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors are very thankful to the reviewer for his/her valuable and constructive comments. This work is supported by Indian Institute of Technology (Indian School of Mines), Dhanbad under grant IIT(ISM) JRF-2019.

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Authors and Affiliations

Authors

Contributions

Randhir Kumar Verma: Investigation, Validation, Writing-original draft, Conceptualization. Akhilesh Prasad: Methodology, Formal analysis, Supervision, Writing-review & editing.

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Correspondence to Randhir Kumar Verma.

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The authors declare no conflict of interest.

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Communicated by Franz Luef.

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This article is part of the topical collection “Harmonic Analysis and Operator Theory” edited by H. Turgay Kaptanoglu, Aurelian Gheondea and Serap Oztop.

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Prasad, A., Verma, R.K. Weyl Transforms Associated with Dunkl Wavelet Transform. Complex Anal. Oper. Theory 17, 110 (2023). https://doi.org/10.1007/s11785-023-01414-z

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