  We generalize the classic Fourier transform operator $\mathcal{F}_{p}$ by
using the Henstock-Kurzweil integral theory. It is shown that the operator
equals the $HK$-Fourier transform on a dense subspace of $\mathcal{ L}^p$,
$1<p\leq 2$. In particular, a theoretical scope of this representation is
raised to approximate numerically the Fourier transform of functions on the
mentioned subspace. Besides, we show differentiability of the Fourier transform
function $\mathcal{F}_{p}(f)$ under more general conditions than in Lebesgue's
theory.
