Real Paley-Wiener type theorems for the Dunkl transform on \({\mathcal{S}}^{\prime}(\mathbb{R}^{d})\) (Q1037560)

The authors generalize the results of \textit{P. K. Banerji, S. K. Al-Omari} and \textit{L. Debnath} [Integral Transforms Spec. Funct. 17, No. 11, 759--768 (2006; Zbl 1131.42003)] for the Dunkl transform on \(\mathbb R^d\). More precisely they prove real Paley-Wiener type theorems for the Dunkl transform \(\mathcal F_{D}\) on the space \({\mathcal{S}}^{\prime}(\mathbb{R}^{d})\) of tempered distributions. Let \(T\in S^{\prime}(\mathbb R^{d})\) and \(\Delta _{\kappa }\) the Dunkl Laplacian operator. First, we establish that the support of \(\mathcal F_{D }(T)\) is included in the Euclidean ball \(\bar{\text{B}}(0,M)=\{x\in\mathbb{R}^{d},\, \| x\| \leq M\} , M>0\), if and only if for all \(R>M\) we have \(\lim_{n\rightarrow +\infty } R ^{ - 2n }\Delta^n_{\kappa} T=0\) in \(S^{\prime}(\mathbb R^d)\). Second, we prove that the support of \(\mathcal F_{D }(T)\) is included in \(\mathbb R^d\setminus B(0,M), M>0\), if and only if for all \(R<M\), we have \(\lim_{n\rightarrow +\infty } R ^{2n} \mathcal F^{-1}_D(\| y\| ^{-2n}\mathcal F_{D }(T))=0\) in \(S^{\prime}(\mathbb R^d)\). Finally, we study real Paley-Wiener theorems associated with \({\mathcal{C}}^{\infty}\)-slowly increasing function.