Fourier transform inversion in the Alexiewicz norm

Abstract

If $f\in L^1(\mathbb{R})$ it is proved that ${\lim }_{S\to \infty }\Vert f-f\ast D_S\Vert =0$, where $D_S(x)=\sin (Sx)/(\pi x)$ is the Dirichlet kernel and $\Vert f\Vert ={\sup }_{\alpha <\beta }|{\int }_{\alpha }^{\beta }f(x)dx|$ is the Alexiewicz norm. This gives a symmetric inversion of the Fourier transform on the real line. An asymmetric inversion is also proved. The results also hold for a measure given by $dF$ where $F$ is a continuous function of bounded variation. Such measures need not be absolutely continuous with respect to Lebesgue measure. An example shows there is $f\in L^1(\mathbb{R})$ such that ${\lim }_{S\to \infty }\Vert f-f\ast D_S{\Vert }_1\ne 0$.