  In this paper we present generalisations of Paley-Wiener type theorems to
Mellin and (Laplace-)Fourier transforms of rapidly decreasing smooth functions
with positive support and log-polyhomogeneous asymptotic expansion at zero.
This article is based on the thesis \cite{CdC} and uses results borrowed from
\cite{RS}, \cite{RS81}, \cite{P79}, \cite{FGD}, \cite{P79}, \cite{SZ}.
