10.57647/

Generalization of Titchmarsh’s Theorem for the Dunkl Transform in the Space Lp(Rd; wl(x)dx)

PDF
  • salah El ouadih1,
  • Radouan Daher1,
  1. Department of Mathematics, Faculty of Sciences A¨ın Chock,University Hassan II, Casablanca, Morocco.

Received: 27-07-2016

Revised: 12-10-2016

Accepted: 28-11-2016

Published in Issue 28-07-2025

Copyright (c) 2025 salah El ouadih, Radouan Daher (Author)

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

El ouadih, salah, & Daher, R. (2025). Generalization of Titchmarsh’s Theorem for the Dunkl Transform in the Space Lp(Rd; wl(x)dx). International Journal of Mathematical Modelling & Computations, 6(4). https://doi.org/10.57647/
Crossref
Scopus
Google Scholar
Europe PMC

Abstract

Using a generalized spherical mean operator, we obtain a generalization of Titchmarsh’s theorem for the Dunkl transform for functions satisfying the (φ; p)-Dunkl Lipschitz condition in the space Lp(Rd; wl(x)dx), 1 < p ⩽ 2, where wl is a weight function invariant under the action of an associated reflection group. 

Keywords

  • Dunkl transform,
  • Dunkl kernel,
  • Generalized spherical mean operator
PDF

References

  1. [1] V. A. Abilov and F. V. Abilova, Approximation of functions by Fourier-Bessel sums. Izv. Vyssh.
  2. Uchebn. Zaved, 8 (2001) 3-9.
  3. [2] E. S. Belkina and S. S. Platonov, Equivalence of K-functionals and modulus of smoothness constructed by generalized Dunkl translations. Izv. Vyssh. Uchebn. Zaved, 8 (2008) 3-15.
  4. [3] M. F. E. de Jeu, The Dunkl transform. Inv. Math, 113 (1993) 147-162.S. El ouadih & R. Daher/ IJM2C, 6 - 4 (2016) 261-267. 267
  5. [4] C. F. Dunkl, Differential- difference operators associated to reflection groups. Trans. Am. Math Soc,
  6. 311 (1989) 167-183.
  7. [5] C. F. Dunkl, Hankel transforms associated to finite reflection groups. In: Proceedings of Special
  8. Session on Hypergeometric Functions in Domains of Positivity. Jack Polynomials and Applications,
  9. Contemp, Math, 138 (1992) 123-138.
  10. [6] C. F. Dunkl, Integral Kernels with reflection group invariance. Canad. J. Math, 43 (1991) 1213-1227.
  11. [7] C. F. Dunkl and Y. Xu, Orthogonal Polynomials of Several Variables. Encyclopedia of Mathematics
  12. and its Applications, Cambridge University Press, Cambridge, 81 (2001).
  13. [8] M. Maslouhi, An analog of Titchmarshs Theorem for the Dunkl transform. Integral Transform Spec.
  14. Funct, 21 (10) (2010) 771-778.
  15. [9] M. Rsler and M. Voit, Markov processes with Dunkl operators. Adv. Appl. Math, 21 (1998) 575-643.
  16. [10] S. Thangavelu and Y. Xu, Convolution operator and maximal function for Dunkl transform. J. Anal.
  17. Math, 97 (2005) 25-56.
  18. [11] E. C. Titchmarsh, Introduction to the theory of Fourier integrals. Claredon, oxford, (1948),
  19. Komkniga. Moxow, (2005).
  20. [12] E. C. Titchmarsh, Introduction to the theory of Fourier integrals, Clarendon Press, Oxford, (1937).
  21. [13] K. Trim` eche, Paley-Wiener theorems for the Dunkl transform and Dunkl transform operators. Integral Transf. Spec. Funct, 13 (2002) 17-38.

Most read articles by the same author(s)