Fourier analysis with generalized integration

Abstract

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.