Does there exist a non-negative valued compactly supported function such that its Fourier transform only vanishes at a given point?

Question

My question is as follows: Given $t_0\in\mathbb{R}$. Does there exist a non-negative valued compactly supported function $f\in L^1(\mathbb{R})$ such that its Fourier transform, $\widehat f\left( t \right)$, only vanishes at $t_0$, $-t_0$?

Answer 1

Assuming that $f(x)$ is supported on $[-1,1]$ we have that $\widehat{f}(s)$ is an entire function of exponential type $1$ by Paley-Wiener theorem. If $\widehat{f}$ is not allowed to have any complex root out of $\pm t_0$, then $$g(s)=\frac{\widehat{f}(s)}{s^2-t_0^2}$$ is an entire function of exponential type $1$ with no roots, hence $g(s)=\exp(as+b)$ by Hadamard's theorem. In this case, however, by considering the inverse Fourier transform we do not get a real valued function.