I have a basic question which I am confused about. If I have two functions $f(x)$, and $g(x)$, where $g(x)$ has a compact support say $[-M,M]$. Can I always say that I can just consider the integral over $[-M,M]$. Namely,
$$ f*g(x)= \int_{-M}^{M}f(x-y)g(y)dy= \int_{-M}^{M}f(y)g(x-y)dy $$ I am worried that what if $f$ blows up outside the support of $g(x)$. I know that if $f(x)$ is bounded, then there is no problem what that statement. Thanks!
The first equality is correct but the second one is not. The last integral should be changed to $\int_{x-M}^{x+M} f(y)g(x-y)\, dy$.