Fourier analysis on the Heisenberg group.

We obtain a usable characterization of the (group) Fourier transform of (H(n)) (Schwartz space on the Heisenberg group). The characterization involves writing elements of [Formula: see text] as asymptotic series in Planck's constant. In the process, we derive a new "discrete" version of spherical harmonics, and elucidate the theory of group contractions. We give an application to Hardy space theory.