On pointwise inversion of the Fourier transform of $BV_{0}$ functions

Abstract

‎Using a Riemann-Lebesgue lemma for the Fourier transform over the class of‎ ‎bounded variation functions that vanish at infinity‎, ‎we prove the‎ ‎Dirichlet--Jordan theorem for functions on this class‎. ‎Our proof is in the‎ ‎Henstock--Kurzweil integral context and is different to that of‎ ‎Riesz-Livingston [Amer‎. ‎Math‎. ‎Monthly 62 (1955)‎, ‎434--437]‎. ‎As consequence‎, ‎we obtain the Dirichlet--Jordan theorem‎ ‎for functions in the intersection of the spaces of bounded variation‎ ‎functions and of Henstock--Kurzweil integrable functions‎. ‎In this‎ ‎intersection there exist functions in $L^{2}(\mathbb{R})\backslash L(\mathbb{%‎ ‎R}).$‎