In dimension one, it is well known that $\mathcal{F}\chi_{(-1,1)}=\frac{\sin{x}}{x}$. This implies, in particular, that $\frac{\sin{x}}{x}$ is a definite positive function. I wonder if a similar result holds in dimension three. Moreover, it would be nice to actually get strictly positive definiteness. So my question is:

**Is it true that the function $f:\mathbb{R}^3\to\mathbb{R}$ given by $f(x)=\frac{\sin{|x|}}{|x|}$ is the Fourier transform of a positive (or at least non-negative) function?**