# Convolution with compact support function

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 $$\int_{x-M}^{x+M} f(y)g(x-y)\, dy$$.
• Oh I see. Thanks! I am wondering whether the first equality always holds. What if $f(x)$ blows up near the compact support of $g(x)$ – Demha Feb 4 '19 at 18:00
• Just to reword what I mean. In order to use $f*g(x)= \int_{-M}^{M}f(x-y)g(y)dy= \int_{-M}^{M}f(y)g(x-y)dy$, I need to make sure that $f(x)$ is bounded right? Thanks again! – Demha Feb 4 '19 at 18:20