Doubly invariant subspaces of the Besicovitch space

TL;DR

This paper extends Wiener’s classical characterization of shift-invariant subspaces to the Besicovitch space of almost periodic functions, providing a new understanding of their structure.

Contribution

It offers an analogue of Wiener’s theorem for the Besicovitch Hilbert space, identifying doubly invariant subspaces in this context.

Findings

Characterization of doubly invariant subspaces in the Besicovitch space

Extension of classical results to almost periodic functions

New insights into the structure of invariant subspaces

Abstract

A classical result of Norbert Wiener characterises doubly shift-invariant subspaces for square integrable functions on the unit circle with respect to a finite positive Borel measure $\mu$, as being the ranges of the multiplication maps corresponding to the characteristic functions of $\mu$-measurable subsets of the unit circle. An analogue of this result is given for the Besicovitch Hilbert space of `square integrable almost periodic functions'.