In this paper we study how zeros of the Fourier transform of a function $f: \mathbb{Z}_p^d \to \mathbb{C}$ are related to the structure of the function itself. In particular, we introduce a notion of bandwidth of such functions and discuss its connection with the decomposition of this function into wavelets. Connections of these concepts with the tomography principle and the Nyquist-Shannon sampling theorem are explored. We examine a variety of cases such as when the Fourier transform of the characteristic function of a set $E$ vanishes on specific sets of points, affine subspaces, and algebraic curves. In each of these cases, we prove properties such as equidistribution of $E$ across various surfaces and bounds on the size of $E$. We also establish a finite field Heisenberg uncertainty principle for sets that relates their bandwidth dimension and spatial dimension.