The Fourier transform with Henstock--Kurzweil and continuous primitive integrals

TL;DR

This paper extends the Fourier transform to Henstock--Kurzweil integrable functions and distributions, establishing new properties, inversion formulas, and convolution results within this generalized integral framework.

Contribution

It introduces a Fourier transform framework for Henstock--Kurzweil integrable functions and distributions, including inversion, convolution, and bounded variation conditions.

Findings

Fourier transform is the second distributional derivative of a Hölder continuous function.

Isometric isomorphism between Fourier transforms and the completion of Henstock--Kurzweil integrable functions.

Pointwise inversion of Fourier transform for functions in L^p, 1<p<∞.

Abstract

For each $f\!:\!\mathbb{R}\to\mathbb{C}$ that is Henstock--Kurzweil integrable on the real line, or is a distribution in the completion of the space of Henstock--Kurzweil integrable functions in the Alexiewicz norm, it is shown that the Fourier transform is the second distributional derivative of a H\"older continuous function. The space of such Fourier transforms is isometrically isomorphic to the completion of the Henstock--Kurzweil integrable functions. There is an exchange theorem, inversion in norm and convolution results. Sufficient conditions are given for an $L^1$ function to have a Fourier transform that is of bounded variation. Pointwise inversion of the Fourier transform is proved for functions in $L^p$ spaces for $1<p<\infty$. The exchange theorem is used to evaluate an integral that does not appear in published tables.