On Pointwise Inversion of the Fourier Transform of Bv0 Functions

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(R)\L(R).