Besicovitch almost periodic function

Summary

A Besicovitch almost periodic function is a locally integrable function on the real line with the following property. It can be approximated in a mean sense by trigonometric polynomials with real frequencies. The mean is the upper average of $\abs{f}^p$ over long intervals $[-T, T]$. This class, introduced by Abram Besicovitch, is much larger than the class of Bohr almost periodic functions. For $p = 2$ it carries a complete Fourier theory with a Parseval identity, and it is used for signals with discrete but irregular spectrum.