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

SIGMA 4 (2008), 092, 14 pages      arXiv:0812.4666
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

We consider the Dunkl intertwining operator Vα and its dual tVα, we define and study the Dunkl Sonine operator and its dual on R. Next, we introduce complex powers of the Dunkl Laplacian Δα and establish inversion formulas for the Dunkl Sonine operator Sα,β and its dual tSα,β. Also, we give a Plancherel formula for the operator tSα,β.

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

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