Fourier transform inversion: Bounded variation, polynomial growth, Henstock--Stieltjes integration

TL;DR

This paper establishes Fourier transform inversion theorems for functions of bounded variation with polynomial growth, using Henstock--Stieltjes integration, extending classical results to broader function classes and distributional settings.

Contribution

It introduces Fourier inversion theorems for functions of bounded variation with polynomial growth using Henstock--Stieltjes integral, including distributional cases and principal value considerations.

Findings

Proves pointwise and distributional Fourier inversion theorems for specific function classes.

Develops inversion formulas using Henstock--Stieltjes integral and differential equations.

Extends classical Fourier analysis results to functions with polynomial growth and regulated functions.

Abstract

In this paper we prove pointwise and distributional Fourier transform inversion theorems for functions on the real line that are locally of bounded variation, while in a neighbourhood of infinity are Lebesgue integrable or have polynomial growth. We also allow the Fourier transform to exist in the principal value sense. A function is called regulated if it has a left limit and a right limit at each point. The main inversion theorem is obtained by solving the differential equation $df(t)-i\omega f(t)=g(t)$ for a regulated function $f$, where $\omega$ is a complex number with positive imaginary part. This is done using the Henstock--Stieltjes integral. This is an integral defined with Riemann sums and a gauge. Some variants of the integration by parts formula are also proved for this integral. When the function is of polynomial growth its Fourier transform exists in a distributional sense, although the inversion formula only involves integration of functions and returns pointwise values.