The double Fourier transform of non-Lebesgue integrable functions of bounded Hardy-Krause variation (Q6111385)

Using a new integral, an improper Kurzweil-Henstock integral, the authors define the Fourier transform on \(\mathbb R^2\) for a class of functions that are not necessarily Lebesgue integrable. For this class of functions, an analogue of the Riemann-Lebesgue lemma is proved. For a subclass of functions, which is characterized by the bounded variation functions in the sense of Hardy-Krause, pointwise continuity of the Fourier transform is established. Analogues of some other properties of the classical Fourier transform on \(L^p\) for \(1< p \leq 2\) are also investigated.