The Range of the Spectral Projection Associated with the Dunkl Laplacian

For s∈ℝ, denote by Pksf the "projection"of a function f in Dℝd into the eigenspaces of the Dunkl Laplacian Δk corresponding to the eigenvalue -s2. The parameter k comes from Dunkl's theory of differential-difference operators. We shall characterize the range of Pks on the space of functions f∈Dℝd supported inside the closed ball BO,R¯. As an application, we provide a spectral version of the Paley-Wiener theorem for the Dunkl transform.