Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 4 (2008), 067, 22 pages      arXiv:0809.5021      https://doi.org/10.3842/SIGMA.2008.067
Contribution to the Special Issue on Dunkl Operators and Related Topics

Inversion Formulas for the Dunkl Intertwining Operator and Its Dual on Spaces of Functions and Distributions

Khalifa Trimèche
Faculty of Sciences of Tunis, Department of Mathematics, 1060 Tunis, Tunisia

Received May 13, 2008, in final form September 16, 2008; Published online September 29, 2008

Abstract
In this paper we prove inversion formulas for the Dunkl intertwining operator Vk and for its dual tVk and we deduce the expression of the representing distributions of the inverse operators Vk−1 and tVk−1, and we give some applications.

Key words: inversion formulas; Dunkl intertwining operator; dual Dunkl intertwining operator.

pdf (239 kb)   ps (209 kb)   tex (19 kb)

References

  1. Chazarain J., Piriou A., Introduction to the theory of linear partial differential equations, North-Holland Publishing Co., Amsterdam - New York, 1982.
  2. van Diejen J.F., Confluent hypergeometric orthogonal polynomials related to the rational quantum Calogero system with harmonic confinement, Comm. Math. Phys. 188 (1997), 467-497, q-alg/9609032.
  3. Dunkl, C.F., Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), 167-183.
  4. Dunkl C.F., Integral kernels with reflection group invariance, Canad. J. Math. 43 (1991), 1213-1227.
  5. Dunkl C.F., Hankel transform associated to finite reflection groups, Contemp. Math. 138 (1992), 123-138.
  6. Heckman G.J., An elementary approach to the hypergeometric shift operators of Opdam, Invent. Math. 103 (1991), 341-350.
  7. Humphreys J.E., Reflection groups and Coxeter groups, Cambridge University Press, Cambridge, 1990.
  8. Hikami K., Dunkl operators formalism for quantum many-body problems associated with classical root systems, J. Phys. Soc. Japan 65 (1996), 394-401.
  9. de Jeu M.F.E., The Dunkl transform, Invent. Math. 113 (1993), 147-162.
  10. de Jeu M.F.E., Paley-Wiener theorems for the Dunkl transform, Trans. Amer. Math. Soc. 258 (2006), 4225-4250, math.CA/0404439.
  11. Kakei S., Common algebraic structure for the Calogero-Sutherland models, J. Phys. A: Math. Gen. 29 (1996), L619-L624, solv-int/9608009.
  12. Lapointe M., Vinet L., Exact operator solution of the Calogero-Sutherland model, Comm. Math. Phys. 178 (1996), 425-452, q-alg/9509003.
  13. Rösler M., Voit M., Markov processes related with Dunkl operators, Adv. in Appl. Math. 21 (1998), 575-643.
  14. Rösler M., Positivity of Dunkl's intertwining operator, Duke. Math. J. 98 (1999), 445-463, q-alg/9710029.
  15. Trimèche K., The Dunkl intertwining operator on spaces of functions and distributions and integral representation of its dual, Integral Transform. Spec. Funct. 12 (2001), 349-374.
  16. Trimèche K., Generalized harmonic analysis and wavelet packets, Gordon and Breach Science Publishers, Amsterdam, 2001.
  17. Trimèche K., Paley-Wiener theorems for the Dunkl transform and Dunkl translation operators, Integral Transform. Spec. Funct. 13 (2002), 17-38.

Previous article   Next article   Contents of Volume 4 (2008)