Quick way to compute $\int_{0}^{\infty} f(x) g(x+y)\, dx$ for $y \in [0,\infty)$

Question

When we want to compute: $$ h(y) = \int_0^y f(x) g(y-x)\, dx, $$ for $y \in [0,\infty)$ we can make use of a Fast Fourrier Transform to quickly compute these integrals. I have a similar problem where I want to compute: $$ h(y) = \int_{0}^{\infty} f(x) g(x+y)\, dx $$ for all $y \in [0,\infty)$. Is there a similar trick that can be used here?