I am looking for conditions on a function $f(x)$ for its Fourier transform $f(k)$ to not be always $\geq 0$ for all $k$; i.e. a condition on $f(x)$ such that $f(k)<0$ for some k, where $f(k)$ is the Fourier transform of $f(x)$.
In my case the function should be a kernel, so conditions compatible with that ($f(x)$ needs to be positive, even and normalized). I have noticed that if I take $f(x)\sim e^{-|x|^\alpha}$, for $\alpha\leq 2$, its Fourier transform is always positive, instead for $\alpha>2$, it is negative for some $k$.
I was therefore thinking it might have something to do with the ``sharpness" (1st or 2nd derivative) of the decrease (try plotting $f(x)\sim e^{-|x|^\alpha}$ for various $\alpha$ to see what I mean), but I couldn't find much. At support of my thesis there is this (cambridge.org/core/services/aop-cambridge-core/content/view/D232B9DEB4819E4D2FF17E8162B8B066/S0004972700047511a.pdf/on_positivity_of_fourier_transforms.pdf) paper showing that convex functions have positive Fourier transform, but it doesn't really sove my problem.
Do you have any idea of what a condition might be and how to prove it? Do you have references you think might help me?