The Paley–Wiener theorem for certain nilpotent Lie groupsJean

Abstract

We generalize the classical Paley-Wiener theorem to special types of connected, simply connected, nilpotent Lie groups: First we consider nilpotent Lie groups whose Lie algebra admits an ideal which is a polarization for a dense subset of generic linear forms on the Lie algebra. Then we consider nilpotent Lie groups such that the co-adjoint orbits of all the elements of a dense subset of the dual of the Lie algebra g * are flat.

Paley‐Wiener estimates for the Heisenberg group

“…Some comments on the relations between results in this paper and results in the literature are in order. The study of bandlimited functions and Paley-Wiener-type results recently has been an active branch of research, for noncommutative groups [15,25,18,16,2], and for symmetric spaces, see e.g. [20,14,22].…”

“…See e.g. [25,18,2] for such results for certain classes of nilpotent groups including the Heisenberg group, and [15,16] for more general cases. By contrast, we consider an "inverse" version: For a function f with compactly supported Fourier transform (the latter being defined by spectral theory of the sub-Laplacian), we study complex-analytic properties of f .…”

Paley-Wiener theorems with respect to the spectral parameter

“…There is no answer so far for the Gelfand pair (U(n) ⋉ H n , U(n)), where H n is the 2n + 1-dimensional Heisenberg group. Even though some attempts have been made to address this Paley-Wiener theorem for the Heisenberg group, [9,24,25,23]. The Fourier analysis for symmetric spaces of noncompact type is well understood by the work of Helgason and Gangolli, [10,14].…”