\begin{definition}[Definition:Fourier Transform]
The '''Fourier transform''' of a Lebesgue integrable function $f: \R^N \to \C$ is the function $\map \FF f: \R^N \to \C$ given by:
:$\ds \map {\map \FF f} {\mathbf s} := \int_{\R^N} \map f {\mathbf x} e^{-2 \pi i \mathbf x \cdot s} \rd \mathbf x$
for $\mathbf s \in \R^N$.
Here, the product $\mathbf x \cdot \mathbf s$ in the exponential is the dot product of the vectors $\mathbf x$ and $\mathbf s$.
\end{definition}