Real Paley-Wiener type theorems for the Dunkl transform on ${\mathcal{S}}'(\mathbb{R}^{d})$

Abstract

We prove real Paley-Wiener type theorems for the Dunkl transform ℱD on the space \({\mathcal{S}}'(\mathbb{R}^{d})\) of tempered distributions. Let T∈S′(ℝd) and Δκ the Dunkl Laplacian operator. First, we establish that the support of ℱD(T) is included in the Euclidean ball \(\bar{\mathrm{B}}(0,M)=\{x\in\mathbb{R}^{d},\ \Vert x\Vert \leq M\}\) , M>0, if and only if for all R>M we have lim n→+∞R−2nΔκnT=0 in S′(ℝd). Second, we prove that the support of ℱD(T) is included in ℝd∖B(0,M), M>0, if and only if for all R<M, we have lim n→+∞R2n ℱD−1(‖y‖−2nℱD(T))=0 in S′(ℝd). Finally, we study real Paley-Wiener theorems associated with \({\mathcal{C}}^{\infty}\) -slowly increasing function.