The support of the convolution product of two functions

Question

Suppose that we have a fuction $f$ supported in a domain $\Omega_1$ and another function $g$ supported in $\Omega_2$, do i have $f*g$ supported in $ \Omega_1 + \Omega_2$? I read it in a book and i still not sure, please sketsh the proof.

Answer 1

In order for $$(f * g)(t) = \int_\Bbb R f(\tau)g(t - \tau)\,d\tau$$ to be non-zero, we must have that somewhere the integrand is non-zero. That is, there is some $\tau$ such that $f(\tau)g(t - \tau)$ is not zero. This requires $\tau \in \Omega_1$ and $t - \tau \in \Omega_2$. So:

$$t \in \Omega_2 + \tau \subseteq \Omega_2 + \Omega_1$$

So yes, the support of $f*g$ is contained in $\Omega_1 + \Omega_2$.

If both $f$ and $g$ are positive everywhere, then you can even show equality. But that does not hold in general.