We prove several Paley--Wiener-type theorems related to the spherical
transform on the Gelfand pair (Hn​⋊U(n),U(n)), where Hn​ is
the 2n+1-dimensional Heisenberg group.
Adopting the standard realization of the Gelfand spectrum as the Heisenberg
fan in R2, we prove that spherical transforms of U(n)--invariant functions and distributions with compact support in Hn​
admit unique entire extensions to C2, 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