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

For each $f\text{\hspace{0.17em}⁣}:\text{\hspace{0.17em}⁣}\mathbb{R}\to \mathbb{C}$ 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 $1<p<\mathrm{\infty }$. The exchange theorem is used to evaluate an integral that does not appear in published tables.