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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 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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