We use the folling conventions for the Fourier transforms in space and time. The temporal Fourier transform: \begin{equation} \widehat{v}(\omega) = F_tv = \frac{1}{\sqrt{2\pi}}\int v(t)\exp(\imath\omega t)\mathrm{d}t, \end{equation} \begin{equation} v(t) = F_t^{-1}\widehat{v} = \frac{1}{\sqrt{2\pi}}\int \widehat{v}(\omega)\exp(-\imath\omega t)\mathrm{d}\omega \end{equation} The spatial Fourier transform \begin{equation} \widehat{v}(\xi) = F_xv = \frac{1}{(2\pi)^{n/2}}\int v(x)\exp(\imath\xi\cdot x)\mathrm{d}x, \end{equation} \begin{equation} v(x) = F_x^{-1}\widehat{v}= \frac{1}{(2\pi)^{n/2}}\int \widehat{v}(\xi)\exp(-\imath\xi\cdot x)\mathrm{d}\xi, \end{equation} Parseval's theorem ensures that the Fourier transform of a square integrable is also square integrable: \begin{equation} \int |v(t)|^2\mathrm{d}t = \int |\widehat{v}(\omega)|^2\mathrm{d}\omega. \end{equation} The Fourier transform is defined for distributions as well, in particular \begin{equation} \delta(t) = \frac{1}{\sqrt{2\pi}}\int \exp(-\imath\omega t)\mathrm{d}\omega. \end{equation}
Some other usefull relations:
differentation : \begin{equation*} F_t v'(\omega) = (\imath\omega)F_tv(\omega), \end{equation*}
translation : \begin{equation*} F_t v(\cdot + \tau)(\omega) = e^{\imath\omega\tau}F_tv(\omega), \end{equation*}
convolution : \begin{equation*} F_t (u\star v)(\omega) = \left(F_tu\right)\left(F_tv\right)(\omega). \end{equation*}
correlation
: \begin{equation*}
F_t (u * v)(\omega) = \overline{\left(F_tu\right)}\left(F_tv\right)(\omega),
\end{equation*}
where
Finally, recall that we may uniquely reconstruct a bandlimited function