# Positive-definiteness of radial sinc function in three dimensions

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?

• Would you say what $J_\nu$ is (just a keyword, searchable)? – YCor Nov 17 '18 at 17:48
• I just mean that I would be able to understand the post (by yourself) you link at, if you give the name/definition of the function $J_\nu$ ("Bessel function" or so). – YCor Nov 17 '18 at 18:20
• @YCor Oh! I see. I thought it’d be enough that formula (3) there has the required $\frac{\sin|x|}{|x|}$. – Francois Ziegler Nov 17 '18 at 18:23