LAUSR

Abstract In the classical Fourier analysis, the representation of the double Fourier transform as the integral of f ( x , y ) exp ( − i ⟨ ( x , y ) , ( s 1 , s 2 ) ⟩ f(x,y)\exp(-i\langle(x,y),(s_{1},s_{2})\rangle is usually defined from the Lebesgue integral. Using an improper Kurzweil–Henstock integral, we obtain a similar representation of the Fourier transform of non-Lebesgue integrable functions on R 2 \mathbb{R}^{2} . We prove the Riemann–Lebesgue lemma and the pointwise continuity for the classical Fourier transform on a subspace of non-Lebesgue integrable functions which is characterized by the bounded variation functions in the sense of Hardy–Krause. Moreover, we generalize some properties of the classical Fourier transform defined on L p ⁢ ( R 2 ) L^{p}(\mathbb{R}^{2}) , where 1 < p ≤ 2 1