I want to find two functions $f_1, f_2 : \mathbb{R} \rightarrow \mathbb{R}$ which are not zero almost everywhere, but $f_1 * f_2$ is a zero function.
My first idea is to try function with disjoint support. So I choose two simple functions with disjoint compact supports $$f_1 = \chi_{[0,1]}, f_2 = \chi_{[2,3]}.$$ Then $$f_1 * f_2 (x) = \int_0^1 \chi_{[2,3]}(x-y) dy.$$ When $x$ is such that $x-y \in [0,1]$, the convolution wont be zero there.
Next I try a simple function with function that integrate to zero on a finite interval like $f(x) = x - 3$ on $[2,4]$ and zero outside. The convolution still not zero.
So I am not sure what type of functions I should looks for.
Any suggestion ?
Kavi had the right idea, but smoothness is unnecessary. The functions \begin{eqnarray} f(x) &=& \frac{\sin(x)}{x} \\ g(x) &=& \frac{\sin(x)}{x}\cos(3x) \end{eqnarray} have Fourier transforms with finite and disjoint support, so their convolution vanishes identically.