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