# Università degli Studi di Genova

## Paley-Wiener Theorems for the 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 (H_n\rtimes 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 R^2, we prove that spherical transforms of U(n)-invariant functions and distributions with compact support in H_n admit a unique entire extension to C^2, and we find a 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. This requires a preliminary analysis of spherical transforms of U(n)-invariant tempered distributions, which are identified as distributions on R^2 supported on the fan which are synthetizable, i.e., vanishing on functions which are zero on the fan.

Keywords:
Fourier trasform, Schwartz space, Paley-Wiener Theorems, Heisenberg group

MSC:
43