I have a question regarding convolution with compact support:
Suppose $f \in L^1(\mathbb{R})$ and $g \in L^p(\mathbb{R})$, and both of them have compact support.
Show that $f*g$ (convolution integral of $f$ and $g$) has compact support.
Kindly advise in proceeding the working.
Thank you.
Best Answer
We define $f \ast g (x)$ to be
$$ f \ast g(x) = \int f(y)g(x-y) dy$$
If this integral is to be nonzero, there must be some overlap between the support of $f$ and the support of $g(x-y)$. What does changing $x$ do to the support of $g(x-y)$? If both $f$ and $g$ are compact support, can you see why taking $x$ large enough will force the supports of the functions in the integral to be disjoint?