  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.
