Suppose that $f$ is a function of the form $$ f(x) = g(x+c)g(x), $$ i.e. $f$ is a product of $g$ with a shifted $g$. If $g$ is a Gaussian, so is $f$ for every $c \in \mathbb R$. In particular, the Fourier transform of $f$ does not vanish.
Does there exist other functions with this property or even a class of functions $g$ such that the Fourier transform of $x \mapsto g(x+c)g(x)$ does not vanish for every $c$?