SIGMA 4 (2008), 092, 14 pages      arXiv:0812.4666      https://doi.org/10.3842/SIGMA.2008.092
Contribution to the Special Issue on Dunkl Operators and Related Topics

Sonine Transform Associated to the Dunkl Kernel on the Real Line

Fethi Soltani
Department of Mathematics, Faculty of Sciences of Tunis, Tunis-El Manar University, 2092 Tunis, Tunisia

Received June 19, 2008, in final form December 19, 2008; Published online December 26, 2008

Abstract
We consider the Dunkl intertwining operator V_α and its dual ^tV_α, we define and study the Dunkl Sonine operator and its dual on R. Next, we introduce complex powers of the Dunkl Laplacian Δ_α and establish inversion formulas for the Dunkl Sonine operator S_α,β and its dual ^tS_α,β. Also, we give a Plancherel formula for the operator ^tS_α,β.

Key words: Dunkl intertwining operator; Dunkl transform; Dunkl Sonine transform; complex powers of the Dunkl Laplacian.

pdf (246 kb)   ps (168 kb)   tex (13 kb)

References

1. Baccar C., Hamadi N.B., Rachdi L.T., Inversion formulas for Riemann-Liouville transform and its dual associated with singular partial differential operators, Int. J. Math. Math. Sci. 2006 (2006), Art. ID 86238, 26 pages.
2. Dunkl C.F., Differential-difference operators associated with reflections groups, Trans. Amer. Math. Soc. 311 (1989), 167-183.
3. Dunkl C.F., Integral kernels with reflection group invariance, Canad. J. Math. 43 (1991), 1213-1227.
4. Dunkl C.F., Hankel transforms associated to finite reflection groups, Contemp. Math. 138 (1992), 123-138.
5. de Jeu M.F.E., The Dunkl transform, Invent. Math. 113 (1993), 147-162.
6. Lapointe L., Vinet L., Exact operator solution of the Calogero-Sutherland model, Comm. Math. Phys. 178 (1996), 425-452, q-alg/9509003.
7. Lebedev N.N., Special functions and their applications, Dover Publications, Inc., New York, 1972.
8. Ludwig D., The Radon transform on Euclidean space, Comm. Pure. App. Math. 23 (1966), 49-81.
9. Nessibi M.M., Rachdi L.T., Trimèche K., Ranges and inversion formulas for spherical mean operator and its dual, J. Math. Anal. Appl. 196 (1995), 861-884.
10. Rosenblum M., Generalized Hermite polynomials and the Bose-like oscillator calculus, in Nonselfadjoint Operators and Related Topics (Beer Sheva, 1992), Oper. Theory Adv. Appl., Vol. 73, Birkhäuser, Basel, 1994, 369-396, math.CA/9307224.
11. Rösler M., Bessel-type signed hypergroups on R, in Probability Measures on Groups and Related Structures, XI (Oberwolfach, 1994), Editors H. Heyer and A. Mukherjea, Oberwolfach, 1994, World Sci. Publ., River Edge, NJ, 1995, 292-304.
12. Rösler M., Generalized Hermite polynomials and the heat equation for Dunkl operators, Comm. Math. Phys. 192 (1998), 519-542, q-alg/9703006.
13. Samko S.G., Hypersingular integrals and their applications, Analytical Methods and Special Functions, Vol. 5, Taylor & Francis, Ltd., London, 2002.
14. Solmon D.C., Asymptotic formulas for the dual Radon transform and applications, Math. Z. 195 (1987), 321-343.
15. Soltani F., Trimèche K., The Dunkl intertwining operator and its dual on R and applications, Preprint, Faculty of Sciences of Tunis, Tunisia, 2000.
16. Stein E.M., Singular integrals and differentiability properties of functions, Princeton University Press, Princeton, 1970.
17. Trimèche K., Transformation intégrale de Weyl et théorème de Paley-Wiener associés à un opérateur différentiel singulier sur (0,¥), J. Math. Pures Appl. (9) 60 (1981), 51-98.
18. 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.
19. Trimèche K., Paley-Wiener theorems for the Dunkl transform and Dunkl translation operators, Integral Transform. Spec. Funct. 13 (2002), 17-38.
20. Xu Y., An integral formula for generalized Gegenbauer polynomials and Jacobi polynomials, Adv. in Appl. Math. 29 (2002), 328-343.