Fourier transform inversion in the Alexiewicz norm

TL;DR

This paper establishes Fourier transform inversion in the Alexiewicz norm for functions in L^1(R), including symmetric and asymmetric versions, and extends results to certain measures, with an example showing limitations of convergence in L^1 norm.

Contribution

It introduces a novel Fourier inversion method using the Alexiewicz norm and extends the theory to measures of bounded variation, providing new insights into convergence properties.

Findings

Convergence of Fourier inversion in Alexiewicz norm for L^1 functions.

Extension of inversion results to measures of bounded variation.

Counterexample showing L^1 convergence may fail.

Abstract

If $f\in L^1({\mathbb R})$ it is proved that $\lim_{S\to\infty}\lVert f-f\ast D_S\rVert=0$, where $D_S(x)=\sin(Sx)/(\pi x)$ is the Dirichlet kernel and $\lVert f\rVert = \sup_{\alpha<\beta}|\int_{\alpha}^{\beta}f(x)\,dx|$ 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 $\lim_{S\to\infty} \rVert f-f\ast D_S\lVert_1\neq 0$.