Spectrum of Functions and Distributions for the Jacobi-Dunkl Transform on {mathbb{R}}

We establish real Paley-Wiener theorems for the Jacobi-Dunkl transform on ${mathbb{R}}$ . More precisely, we characterize the functions in the generalized Schwartz space ${mathcal{S}^{r}_{alpha , beta}(mathbb{R})}$ and in ${L^{p}_{{A}_{alpha , beta}} mathbb{R})}$ whose Jacobi-Dunkl transform has bounded, unbounded, convex and nonconvex support. Finally, we study the spectral problem on the generalized tempered distributions ${mathcal{S}^{'r}_{alpha , beta}(mathbb{R})}$ .