The double Fourier transform of non-Lebesgue integrable functions of bounded Hardy–Krause variation

Abstract

AbstractIn the classical Fourier analysis, the representation of the double Fourier transform as the integral off(x,y)exp(−i⟨(x,y),(s1,s2)⟩f(x,y)\exp(-i\langle(x,y),(s_{1},s_{2})\rangleis 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 onR2\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 onLp⁢(R2)L^{p}(\mathbb{R}^{2}), where1<p≤21<p\leq 2, yielding a generalization of the results obtained by E. Hewitt and K. A. Ross. With our integral, we define the space of integrable functionsKP⁢(R2)\mathrm{KP}(\mathbb{R}^{2})which contains a subspace whose completion is isometrically isomorphic to the space of integrable distributions on the plane as defined by E. Talvila [The continuous primitive integral in the plane,Real Anal. Exchange45(2020), 2, 283–326]. A question arises about the dual space of the new spaceKP⁢(R2)\mathrm{KP}(\mathbb{R}^{2}).