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 xU(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.