  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.
