Fourier transform inversion using an elementary differential equation and a contour integral

TL;DR

This paper presents a simplified proof of the Fourier transform inversion theorem using elementary differential equations and contour integrals, making the proof more accessible and extendable to multiple variables.

Contribution

It introduces an elementary approach to prove Fourier inversion using differential equations and contour integrals, reducing reliance on advanced machinery.

Findings

Proof of Fourier inversion theorem with minimal machinery

Extension of method to multiple variables

Application to Riemann's localization theorem

Abstract

Let $f$ be a function on the real line. The Fourier transform inversion theorem is proved under the assumption that $f$ is absolutely continuous such that $f$ and $f'$ are Lebesgue integrable. A function $g$ is defined by $f'(t)-iwf(t)=g(t)$. This differential equation has a well known integral solution using the Heaviside step function. An elementary calculation with residues is used to write the Heaviside step function as a simple contour integral. The rest of the proof requires elementary manipulation of integrals. Hence, the Fourier transform inversion theorem is proved with very little machinery. With only minor changes the method is also used to prove the inversion theorem for functions of several variables and to prove Riemann's localization theorem.