Fourier Transforms and Vector Fields in Discretized and CurvedSpace

This paper explores the consequences of space discretization for the concepts of Fourier trans-
forms and vector fields. Specifically, I will show that under normal conditions (Δ𝑥 𝜅𝑥 ≪ 1) the
Fourier transform of discrete space has the ”expected” form, I will numerically show that in dis-
cretized space the inverse Fourier transform of a Fourier transform returns approximately the same
image (i.e. F −1 ◦ F ≈ 1), I will also show how different definitions of Fourier transform affect the
output when the ”normal conditions” are violated (i.e. when the conditions approach the Nyquist
limit). The most important result of this paper is however a proof of Sotoke’s and Divergence
theorems in discretized space. The proof of these two theorems is not trivial in discretized space
and requires a new ”boundary aware” definitions of ∇ and ∇×. Lastly, I describe Fourier transform
in curved space.