Find a pair of functions $f,g:\mathbb{R}\to\mathbb{R}$ such that:

- $f$ is smooth and compactly supported (say, on $[0,1]$ but this isn't crucial),
- $g(x)>0$ for all $x\in\mathbb{R}$, $\int g(x)\,dx=1$ (i.e. $g$ is a strictly positive density), and
- $f*g=0$.

If we remove the condition that $g$ is a strictly positive density, then this is possible by choosing $g$ such that its Fourier transform is a sum of point masses (e.g. something like $1+\sin^2 x$) and choosing $f$ so that its Fourier transform has prescribed zeroes at these point masses. But I am not sure if the condition that $g$ is a density changes anything.

A stronger reformulation (due to Paley-Wiener) of this problem is: *Is there a strictly positive density whose Fourier transform has a finite number of prescribed zeroes?*

1more comment