Sonine Transform Associated to the Dunkl Kernel on the Real Line

Abstract

We consider the Dunkl intertwining operator $V_{\alpha }$ and its dual ${}^t{V}_{\alpha }$, we define and study the Dunkl Sonine operator and its dual on $\mathbb{R}$. Next, we introduce complex powers of the Dunkl Laplacian ${\Delta }_{\alpha }$ and establish inversion formulas for the Dunkl Sonine operator $S_{\alpha ,\beta }$ and its dual ${}^t{S}_{\alpha ,\beta }$. Also, we give a Plancherel formula for the operator ${}^t{S}_{\alpha ,\beta }$.

This is a contribution to the Special Issue on Dunkl Operators and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

Cited by in corpus (1)

• Inversion of the Dual Dunkl-Sonine Transform on R Using Dunkl Wavelets