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arXiv · 2501.17118

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

Erik Talvila
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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.

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Erik Talvila. 2025-01-28. The Fourier transform with Henstock--Kurzweil and continuous primitive integrals. https://arxiv.org/abs/2501.17118

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