Wiener measure for Heisenberg group

Fourier transform of a Gaussian measure on the Heisenberg group

An explicit formula is derived for the Fourier transform of a Gaussian measure on the Heisenberg group at the Schrödinger representation. Using this explicit formula, necessary and sufficient conditions are given for the convolution of two Gaussian measures to be a Gaussian measure.

Wiener measure and Donsker’s theorem

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 Cn. If f(z)e 1 4 |z| 2 is a Schwartz class function we show that f is supported in a ball of radius B in Cn if and only if f×μr(z) = 0 for r > B+ |z| for all z ∈ Cn. This is an analogue of Helgason’s support theorem on Euclidean and hyperbolic spaces. When n = 1 w...

Sampling theorems for the Heisenberg group

In the first part of the paper a general notion of sampling expansions for locally compact groups is introduced, and its close relationship to the discretisation problem for generalised wavelet transforms is established. In the second part, attention is focussed on the simply connected nilpotent Heisenberg group H. We derive criteria for the existence of discretisations and sampling expansions ...

Discrete Heisenberg-Weyl Group and Modular Group

It is shown that the generators of two discrete Heisenberg-Weyl groups with irrational rotation numbers θ and −1/θ generate the whole algebra B of bounded operators on L2(R). The natural action of the modular group in B is implied. Applications to dynamical algebras appearing in lattice regularization and some duality principles are discussed. Writing a contribution to a memorial volume one alw...