The Fourier transform is considered as a Henstock-Kurzweil integral. Sufficient conditions are given for the existence of the Fourier transform and necessary and sufficient conditions are given for it to be continuous. The Riemann-Lebesgue lemma fails: Henstock-Kurzweil Fourier transforms can have arbitrarily large point-wise growth. Convolution and inversion theorems are established. An appendix gives sufficient conditions for interchanging repeated Henstock-Kurzweil integrals and gives an estimate on the integral of a product.
"Henstock-Kurzweil Fourier transforms." Illinois J. Math. 46 (4) 1207 - 1226, Winter 2002. https://doi.org/10.1215/ijm/1258138475