Let $f$ be an even continuous function with compact support such that $$ \int f(t)\,\mathrm{d}t=1, $$ and let $g$ be a bounded continuous function such that the convolution $f\star g$ satisfies the following equality $$ (f\star g)(x)=g(x). $$ How to prove that if $g$ has global minimum then $g$ is constant?
I thought about using the Convolution Theorem, but it seems it doesn't work in this case. Maybe, without the requirement that $g$ has global minimum, there is a way to prove that $g$ is linear?