Abstract:
We prove several PaleyWienertype theorems related to the spherical transform
on the Gelfand pair (H_n\rtimes U(n),U(n)), where H_n is the
2n+1dimensional 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 realvariable 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, PaleyWiener Theorems, Heisenberg group
MSC:
43
