The Fourier transform of the Gaussian $f(x) = e^{-\pi x^2}$ is again $f$. Let's assume $a,b \in \mathbb{R}$ and $g_1,g_2$ are defined by $$ g_1(x)=e^{-\pi (x-a)^2}, \ \ \ \ \ \ g_2(x)=e^{-\pi (x-b)^2}, $$ i.e. $g_1$ and $g_2$ are translates of $f$. Is there a way to calculate the Fourier transform of the product of $g_1$ and $g_2$, $\ \widehat{g_1g_2}$?
Remark: In fact I'm interested whether or not the zeros of the Fourier transform of $g_1g_2$ are isolated or not. Of course, this question would be easy to answer if I would have an explicit expression of $\widehat{g_1g_2}$.
Hint: write $g_1g_2=e^{-2\pi\left(x-\frac{a+b}{2}\right)^2+c}$, with $c$ a constant you can find.