On the Estimates of Dunkl Kernels

Abstract

In this paper, we are interested in estimates of the Dunkl kernels on some special sets. We improving results of M.F.E. de Jeu and M. Rösler [6].

This is a preview of subscription content, access via your institution.

References

  1. [1]

    B. Amri, Note on Bessel functions of type AN−1, Integral Transform. Spec. Funct., 25 (2014), 448–461.

    MathSciNet  Article  Google Scholar 

  2. [2]

    B. Amri and N. Demni, Laplace-type integral representations of the generalized Bessel function and of the Dunkl kernel of type B2, Moscow Math. J., 17 (2017), 1–15.

    MathSciNet  Article  Google Scholar 

  3. [3]

    D. Constales, H. De Bie and P. Lian, Explicit formulas for the Dunkl dihedral kernel and the (κ, a)-generalized Fourier kernel, J. Math. Anal. Appl., 460 (2018), 900–926.

    MathSciNet  Article  Google Scholar 

  4. [4]

    H. De Bie and P. Lian, The Dunkl kernel and intertwining operator for dihedral groups, arXiv:2003.01646.

  5. [5]

    M. F. E. de Jeu, The Dunkl transform, Invent. Math., 113 (1993), 147–162.

    MathSciNet  Article  Google Scholar 

  6. [6]

    M. F. E. de Jeu and M. Rösler, Asymptotic analysis for the Dunkl kernel, J. Approx. Theory, 119 (2002), 110–126.

    MathSciNet  Article  Google Scholar 

  7. [7]

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

    MathSciNet  Article  Google Scholar 

  8. [8]

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

    MathSciNet  Article  Google Scholar 

  9. [9]

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

    MathSciNet  Article  Google Scholar 

  10. [10]

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

  11. [11]

    NIST Digital Library of Mathematical Functions (DLMF), http://dlmf.nist.gov.

  12. [12]

    M. Rösler, Positivity of Dunkl’s intertwining operator, Duke Math. J., 98 (1999) 445–463.

    MathSciNet  Article  Google Scholar 

  13. [13]

    M. Rösler, Dunkl operators: theory and applications, in: Orthogonal Polynomials and Special Functions (Leuven, 2002), Lect. Notes Math., vol. 1817, Springer-Verlag (2003), pp. 93–135.

    Google Scholar 

  14. [14]

    E. Ch. Titchmarsh, The Theory of Functions (2nd ed.), Oxford University Press (1939).

  15. [15]

    S. Thangavelu and Y. Xu, Convolution operator and maximal function for the Dunkl transform, J. Anal. Math., 97 (2005), 25–56.

    MathSciNet  Article  Google Scholar 

  16. [16]

    J. F. van Diejen and L. Vinet, Calogero-Moser-Sutherland Models, CRM Series in Mathematical Physics, Springer (2000).

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to B. Amri.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Amri, B., Gasmi, A. On the Estimates of Dunkl Kernels. Anal Math 47, 1–12 (2021). https://doi.org/10.1007/s10476-021-0071-0

Download citation

Key words and phrases

  • reflection group
  • Dunkl operator
  • Dunkl kernel

Mathematics Subject Classification

  • primary 51F15
  • 33C67
  • secondary 34E05