When is the product of a Schwartz function and a smooth function guaranteed to be a Schwartz function?

Question

How can we classify functions $g \in C^\infty \left(\mathbb{R}^d\right)$ such that for all Schwartz functions $f$ on $\mathbb{R}^d$, $f g$ is again Schwartz?

I believe this question could be answered using a Paley-Wiener-like theorem for tempered distributions, but I am struggling to find something like that.