TITLE: Conditions for a nonnegative Fourier transform
QUESTION [5 upvotes]: I was wondering if there was known necessary and sufficient conditions for the result of a Fourier transform to be nonnegative, i.e.:
$$ F(f) \ge 0 $$
where $F$ is the Fourier transform.

REPLY [2 votes]: Since the Fourier transform is invertible, you are asking for necessary and sufficient condition on the function $f$ so that 
$$f(t) = \int_{\mathbb{R}}g(s) e^{i t s} ds$$
for some positive function $g$. The function $f$ must be positive (semi) definite, that is, for every $t_1$, $\ldots$, $t_n$ the matrix
$$(f(t_i-t_j))_{i,h}$$
is Hermitian and positive semidefinite.  See positive-definite functions.
Note that in general $g(s) ds$ will be a positive measure ( that can be singular, like a Dirac measure). So one also needs that $f$ comes from an actual function. 
The equivalent condition by @P. Pet: is sometimes easier to check if it is true. This one may be easier to check that it does not happen. 
Check what happens if  $f(t) = e^{a t}$ for some $a\in \mathbb{R}$. $f$ should be a positive "linear combination" of such functions.