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Let ${\boldsymbol{G}}$ be a connected reductive group defined over anon--Archimedean local field $F$. Put $G={\boldsymbol{G}}(F)$. Let $\theta$ bean $F$--automorphism of ${\boldsymbol{G}}$, and let $\omega$ be a smoothcharacter of $G$. This paper is concerned with the smooth complexrepresentations $\pi$ of $G$ such that $\pi^\theta=\pi\circ\theta$ isisomorphic to $\omega\pi=\omega\otimes\pi$. If $\pi$ is admissible, inparticular 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 transformassociates to a compactly supported locally constant function $f$ on $G$, thefunction $(\pi,A)\mapsto \Theta_\pi^A(f)$ on a suitable Grothendieck group.Here we describe its image (Paley--Wiener theorem), and we reduce thedescription of its kernel (spectral density theorem) to a result on thediscrete part of the theory.

La transform\'ee de Fourier pour les espaces tordus sur un groupe r\'eductif $\mathfrak{p}$-adique I. Le th\'eor\`eme de Paley-Wiener