Wavelet decomposition and bandwidth of functions defined on vector spaces over finite fields

TL;DR

This paper explores the relationship between the zeros of Fourier transforms of functions on finite field vector spaces and their structure, introducing a bandwidth concept linked to wavelet decomposition and establishing a finite field uncertainty principle.

Contribution

It introduces a new notion of bandwidth for functions on finite fields and connects it with wavelet decomposition, Fourier zeros, and a finite field uncertainty principle.

Findings

Zeros of Fourier transforms influence function structure.

Bandwidth relates to wavelet decomposition.

Finite field uncertainty principle established.

Abstract

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.