  Let ${\boldsymbol{G}}$ be a connected reductive group defined over a
non--Archimedean local field $F$. Put $G={\boldsymbol{G}}(F)$. Let $\theta$ be
an $F$--automorphism of ${\boldsymbol{G}}$, and let $\omega$ be a smooth
character of $G$. This paper is concerned with the smooth complex
representations $\pi$ of $G$ such that $\pi^\theta=\pi\circ\theta$ is
isomorphic to $\omega\pi=\omega\otimes\pi$. If $\pi$ is admissible, in
particular irreducible, the choice of an isomorphism $A$ from $\omega\pi$ to
$\pi^\theta$ (and of a Haar measure on $G$) defines a distribution
$\Theta_\pi^A={\rm tr}(\pi\circ A)$ on $G$. The twisted Fourier transform
associates to a compactly supported locally constant function $f$ on $G$, the
function $(\pi,A)\mapsto \Theta_\pi^A(f)$ on a suitable Grothendieck group.
Here we describe its image (Paley--Wiener theorem), and we reduce the
description of its kernel (spectral density theorem) to a result on the
discrete part of the theory.
