m at h . FA ] 1 1 Ju l 2 00 5 Real Paley - Wiener theorems for the Dunkl transform on IR d

In this paper, we establish real Paley-Wiener theorems for the Dunkl transform on IR d. More precisely, we characterize the functions in the Schwartz space S(IR d) and in L 2 k (IR d) whose Dunkl transform has bounded, unbounded, convex and nonconvex support. 1 Introduction In the last few years there has been a great interest to real Paley-Wiener theorems for certain integral transforms, see [15] for an overview references and details for this question. In this paper we consider the Dunkl operators T j , j = 1, ..., d, which are the differential-difference operators introduced by C.F.Dunkl in [5]. These operators are very important in pure Mathematics and in Physics. They provide a useful tool in the study of special functions with root systems (see [6].) C.F.Dunkl in [7] (see also [8]) has studied a Fourier transform F D , called Dunkl transform defined for a regular function f by ∀ x ∈ IR d , F D f (x) = IR d K(−ix, y)f (y)ω k (y)dy, where K(−ix, y) represents the Dunkl kernel and ω k a weight function. The aim purpose of this paper is to prove real Paley-Wiener theorems on the Schwartz space S(IR d) and on L 2 k (IR d). More precisely we consider first the Paley-Wiener spaces 1