I am trying to get the multi-dimensional Fourier transform of the following function.
$\sum_k a_ke^{-\frac{(\bar{w}_k\cdot \bar{x})^2}{\sigma^2}}$,
where $\bar{w}_i, \bar{x} \in \mathbb{R}^n$ and $\sigma$ is a constant. To this end, I think I need to evaluate the following integral and just get a weighted sum.
$\int_{\mathbb{R}^n} e^{-(2\pi i \bar{x} \cdot \epsilon + \frac{(\bar{w} \cdot \bar{x})^2}{\sigma^2} )}d\bar{x}$.
where $\epsilon \in \mathbb{R}^n$ is the Fourier index.
I am able to do this if $\bar{x}, \bar{w} \in \mathbb{R}$. But I am stuck with the n-dimensional case. Any help in solving this would be greatly appreciated.