A spectral Paley-Wiener theorem for the Heisenberg group and a support theorem for the twisted spherical means on Cn

We prove a spectral Paley-Wiener theorem for the Heisenberg group by means of a support theorem for the twisted spherical means on C-n. If f(z)e(1/4)vertical bar z vertical bar(2) is a Schwartz class function we show that f is supported in a ball of radius B in C-n if and only if f x mu(r)(z) = 0 for r > B + vertical bar z vertical bar for all z epsilon C-n. This is an analogue of Helgason's support theorem on Euclidean and hyperbolic spaces. When n = 1 we show that the two conditions f x (z) = mu(r) x f(z) = 0 for r > B + vertical bar z vertical bar imply a Support theorem for a large class of functions with exponential growth. Surprisingly enough,this latter result does not generalize to higher dimensions.