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?*



Vanishing convolution between density and compactly supported function
With your hypotheses above, $\widehat{g}:\mathbb{R}\to\mathbb{R}$ is a uniformly continuous function such that $\displaystyle \lim\_{|\gamma|\to\infty} \widehat{g}(\gamma) = 0$, and $$\widehat{f}(z) := \int f(t)e^{-2\pi itz} dt \hspace{28mm} (z\in\mathbb{C})$$ is an analytic function, for which the set $\{z\in\mathbb{C}: \widehat{f}(z)=0\}$ cannot have accumulation points unless $\widehat{f}=0$ on $\mathbb{C}$. Thus, $f\*g=0$ iff $\widehat{f} \widehat{g}=0$ which implies either $f=0$ or $g=0$.


