Let $f(x)$ be a symmetric function from $\mathbb{R}\to \mathbb{R}$, and $\hat f(k)$ be it's Fourier transform.
What are the necessary and sufficient conditions for $\hat f(k)$ to have at least $n$ roots? I have had a look at some literature on conditions for $\hat f(k)$ to have zero roots (such as this paper). Can someone point to towards helpful literature or results on the existence of at least $n$ roots?
Also, are there any related results for Hankel transforms (for a related 2D problem)?
