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

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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.