On pointwise inversion of the Fourier transform of \(\mathrm{BV}_0\) functions (Q533316)

The Dirichlet-Jordan theorem solves the pointwise inversion of the Fourier transform for Lebesgue integrable and bounded variation functions on \(\mathbb R\). Using a Riemann-Lebesgue lemma for the Fourier transform over the class of bounded variation functions that vanish at infinity, the Dirichlet-Jordan theorem is proved for functions in this class. The proof given in this paper is in the Henstock-Kurzweil integral context and is different to that of Riesz-Livingston. As a consequence, the Dirichlet-Jordan theorem for functions in the intersection of the spaces of bounded variation functions and of Henstock-Kurzweil integrable functions is obtained. In this intersection there exist functions that are square integrable, but not Lebesgue integrable.