Sonine Transform Associated to the Dunkl Kernel on the Real Line
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 $\mathbb{R}$. 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 ${}^tS_{α,β}$.
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