  We prove several Paley--Wiener-type theorems related to the spherical
transform on the Gelfand pair $\big(H_n\rtimes U(n),U(n)\big)$, 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.
