A classical result due to Paley and Wiener characterizes the existence of a nonzero function in $L^{2}(\mathbb{R})$, supported on a half-line, in terms of the decay of its Fourier transform. In this paper, we prove an analogue of this result for Damek–Ricci spaces.

Let G be a p-adic reductive group and let U0 be the unipotent radical of a minimal parabolic subgroup of G. We introduce a Fourier transform defined on the space of smooth Whittaker functions on G which are compactly supported modulo U0. We determine its image. The proof follows the proof of Heiermann for the functions on the group.

During the proof, we establish an inversion formula. This formula allows us to prove that an irreducible smooth representation of G, which has a Whittaker model in the space of smooth Whittaker functions on G which are compactly supported modulo U0, is cuspidal.

This work gave us the opportunity to prepare a framework for the study of harmonic analysis on p-adic reductive symmetric spaces: B-matrices and constant term; a study of wave packets.

Let $X=G/H$ be a reductive symmetric space and $K$ a maximal compact subgroup of $G$. We study Fourier transforms of compactly supported $K$-finite distributions on $X$ and characterize the image of the space of such distributions.

Let G be a real special or general linear group and $\sigma_{0}$ be the transpose-inverse involution. We characterize the image of $f \mapsto {\rm tr}(\pi(f) \pi(\sigma_{0}) )$ for irreducible representations $\pi$ of $G \rtimes \langle \sigma_{0}\rangle$, and K-finite $f \in C_{c}^{\infty}(G)$.