The Fourier transform with Henstock--Kurzweil and continuous primitive integrals
Erik Talvila
TL;DR
This work extends Fourier analysis beyond $L^1$ by defining the Fourier transform for Henstock–Kurzweil integrable functions and their Alexiewicz completion ${\mathcal{A}}_c({\mathbb R})$ as the second distributional derivative of an integral with kernel $v_s(t)=(1- i s t - e^{- i s t})/t^2$. It develops a robust Banach-space framework linking ${\mathcal{A}}_c$, ${\mathcal{B}}_c$, ${\mathcal{C}}_c$, and ${\mathcal{D}}_c$, proves an exchange formula suitable for inversion and convolution, and establishes inversion both in the Alexiewicz norm via summability kernels and pointwise inversion for $1<p<\infty$. Key contributions include isometric isomorphisms among generalized function spaces, detailed conditions yielding bounded variation for Fourier transforms, and concrete examples demonstrating the breadth of behavior possible when moving beyond Lebesgue integrable functions. The results significantly broaden the applicability of Fourier analysis to distributions with no pointwise values, offering rigorous inversion and convolution tools in a generalized function setting with potential applications in analysis and signal processing.
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 Hölder 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.