Distributional {D}unkl transform and {D}unkl convolution operators

Betancor, Jorge J.

Bollettino della Unione matematica italiana, Série 8, 9B (2006) no. 1, pp. 221-245

Résumé

In this paper, that is divided in two parts, we study the distributional Dunkl transform on R. In the first part we investigate the Dunkl transform and the Dunkl convolution operators on tempered distributions. We prove that the tempered distributions defining Dunkl convolution operators on the Schwartz space ƒ are the elements of \(\mathcal{O}'_c\), the space of usual convolution operators on \(S\). In the second part we define the distributional Dunkl transform by employing the kernel method. We introduce Frechet function spaces containing the kernel of the Dunkl transform. In theproof of the properties of the distributional Dunkl transform, defined on the correspoding dual spaces, certain representations of the elements of the dual spaces will play an important role. These representations allows us to simplify, in contrast with the previous and usual methods (see, for instance [7] and [13]), the mentioned proofs. Our new procedure also applies to other distributional integral transforms that had been investigated by other authors (Hankel transforms ([7] and [13]), amongst others).