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?**



Positive-definiteness of radial sinc function in three dimensions
It’s the Fourier transform of the rotation-invariant probability [measure](https://mathoverflow.net/questions/149692/fourier-transform-of-the-unit-sphere) on the unit sphere, and as such is positive definite.


Strict positive definiteness also holds, by the theorem in zu Castell, Filbir and Szwarc ([2005](//ams.org/mathscinet-getitem?mr=2186952)).


