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.