Wavelet decomposition and bandwidth of functions defined on vector spaces over finite fields
Abstract
In this paper we study how zeros of the Fourier transform of a function f: Zpd 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.