Inversion Formula for the Dunkl-Wigner Transform and Compactness Property for the Dunkl-Weyl Transforms
Received:January 21, 2015  Revised:April 27, 2015
Key Words: Dunkl transform   Dunkl-Wigner transform   Dunkl-Weyl transforms   inversion formula   boundedness and compactness
Fund Project:Supported by the DGRST Research Project LR11ES11 and CMCU Program 10G/1503.
 Author Name Affiliation Fethi SOLTANI Department of Mathematics, Faculty of Science, Jazan University, P. O. Box 277, Jazan 45142, Saudi Arabia
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We define and study the Fourier-Wigner transform associated with the Dunkl operators, and we prove for this transform an inversion formula. Next, we introduce and study the Weyl transforms $W_{\sigma}$ associated with the Dunkl operators, where $\sigma$ is a symbol in the Schwartz space $\mathcal{S}(\mathbb{R}^d\times\mathbb{R}^d)$. An integral relation between the precedent Weyl and Wigner transforms is given. At last, we give criteria in terms of $\sigma$ for boundedness and compactness of the transform $W_{\sigma}$.