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 ${\mathbb{R}}^2$, we prove that spherical transforms of $U(n)$–invariant functions and distributions with compact support in $H_n$ admit unique entire extensions to ${\mathbb{C}}^2$, 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.