Paley–Wiener theorems for the $${\text {U}(n)}$$ U ( n ) -spherical transform on the Heisenberg group

Abstract

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