Abstract almost periodicity for group actions on uniform topological spaces

TL;DR

This paper develops a unified framework for understanding almost periodicity of functions, measures, and distributions under group actions on uniform topological spaces, linking various classical concepts.

Contribution

It introduces a comprehensive theory connecting Bohr and Bochner almost periodicity through group actions and uniform structures, unifying previous disparate approaches.

Findings

Unified theory for almost periodicity in Banach spaces, measures, and distributions.

Clarified the relation between Bohr and Bochner almost periodicity.

Demonstrated the framework's applicability to various classical examples.

Abstract

We present a unified theory for the almost periodicity of functions with values in an arbitrary Banach space, measures and distributions via almost periodic elements for the action of a locally compact abelian group on a uniform topological space. We discuss the relation between Bohr and Bochner type almost periodicity, and similar conditions, and how the equivalence among such conditions relates to properties of the group action and the uniformity. We complete the paper by demonstrating how various examples considered earlier all fit in our framework.