The uncertainty principle is a general fact about the Fourier transform on locally compact Abelian groups. However, the precise uncertainty principle depends on the type of group. For example, it is well-known that on the additive group of real numbers $\mathbb{R}$, a function $f$ and its Fourier transform $\hat{f}$ can not both be compactly supported. But this depends on certain specific properties of $\mathbb{R}$. I was wondering if it is possible for a function on a non-Euclidean group and its Fourier transform to both have compact support.
I have the mind to consider the Fourier transform on the p-adic integers $\mathbb{Z}_{p}$ or on the p-adic rationals $\mathbb{Q}_{p}$, but unfortunately, I am not yet familiar with the Fourier analysis on these groups to produce an example or some kind of argument very quickly.
Any help would be appreciated. Thanks.