I am trying to calculate the inverse Fourier transform of a multidimensional (3d) Gaussian translated by a coordinate vector $e_1$.
$ \mathscr{F}^{-1}[T_{-e_{1}}\widehat{G(z)}] = \mathscr{F}^{-1}[e^{-e_{1}^{T}x}\widehat{G(z)}] $
where $e_1=\begin{pmatrix}1 & 0 & 0 \end{pmatrix}$ , $z=\begin{pmatrix}z_1 & z_2 & z_3 \end{pmatrix}$ and $G(z)=(2\pi\sigma^2)^{-d/2}e^{-{(z^2)}/2\sigma^2}$
I know that the Fourier transform of a gaussian is $\widehat{G(z)}=e^{-\sigma^2||z||^2/2}$. What do I do when getting the translate of that function? Do I compute the convolution of the two separate functions $\widehat{G(z)}$ and $ e^{-ie_1^Tz}$?
Tried going for the convolution of the two functions and got this:
$ \dfrac{\mathrm{e}^{\mathrm{i}{\tau}-\frac{{\sigma}^2||x||^2}{2}-\mathrm{i}z}}{\sqrt{2{\pi\sigma^2}}} =e^{-i(x_1-\tau)}G_{\sigma^2}(x) $
Is there anything I am missing? Apologies, I do understand this might be a noobs question..
P.S. To be more specific on the notation used, $T_ah(x)=h(x+a)$.