Paley-Wiener theorems for the U(n)-spherical transform on the Heisenberg group

We prove several Paley-Wiener-type theorems related to the spherical transform on the Gelfand pair (H-n U(n), U(n)), where H-n is the 2n + 1-dimensional Heisenberg group. Adopting the standard realization of the Gelfand spectrum as the Heisenberg fan in R-2, we prove that spherical transforms of U(n)-invariant functions and distributions with compact support in H-n admit unique entire extensions to C-2, and we find real-variable characterizations of such transforms. Next, we characterize the inverse spherical transforms of compactly supported functions and distributions on the fan, giving analogous characterizations.