Fourier transform inversion in the Alexiewicz norm

Abstract. If f P LpRq it is proved that limSÑ8‖f ́ f ̊ DS‖ “ 0, where DSpxq “ sinpSxq{pπxq is the Dirichlet kernel and ‖f‖ “ supαăβ | şβ α fpxq 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 P LpRq such that limSÑ8‖f ́ f ̊ DS‖1‰ 0.