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\item Modulation: $\forall k_0 \in \mathbb{R}: \reallywidehat{e^{2 \pi i x k_0} f(x)} = \hat{f}(k - k_0)$
\item Parseval identity: $\forall f,g \in L^2: \langle f, g \rangle = \langle \hat{f}, \hat{g} \rangle$ where $\langle \cdot , \cdot \rangle$ is the canonical $L^2$ scalar product (see [\cite{stubbe}])
\end{itemize}
\paragraph{Convolution definition and theorem}
Let $f, g \in L^2(\mathbb{R})$, then the \textit{convolution} of $f$ with respect to $g$ is defined as follow:
\begin{equation} \label{convolution}
(f \circledast g)(x) := \int_{-\infty}^{\infty} f(y)g(x - y) dy
\end{equation}
Having listed enough properties of the \textit{Fourier transform}, it's now possible to relate these two concepts by stating the \textit{convolution} theorem: