Real Paley–Wiener theorems and Roe's theorem associated with the Opdam–Cherednik transform

We reconsider Roe's theorem associated with Jacobi–Cherednik operators. We first prove two new real Paley–Wiener theorems for the Opdam–Cherednik transform, characterizing those functions whose transform has support inside or outside intervals. As a corollary, we get a characterization of those functions whose transform has support at the endpoints, which we use to prove our version of Roe's theorem.