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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?

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It’s the Fourier transform of the rotation-invariant probability measure 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).

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  • $\begingroup$ Would you say what $J_\nu$ is (just a keyword, searchable)? $\endgroup$
    – YCor
    Commented Nov 17, 2018 at 17:48
  • $\begingroup$ 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). $\endgroup$
    – YCor
    Commented Nov 17, 2018 at 18:20
  • $\begingroup$ @YCor Oh! I see. I thought it’d be enough that formula (3) there has the required $\frac{\sin|x|}{|x|}$. $\endgroup$
    – Francois Ziegler
    Commented Nov 17, 2018 at 18:23
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    $\begingroup$ It's enough stricto sensu, but I'd be happy to understand the more general result, so as to grasp some intuition. $\endgroup$
    – YCor
    Commented Nov 17, 2018 at 23:04

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