Harald Bohr's splitting theorem

TL;DR

This paper provides a new elementary proof of Harald Bohr's splitting theorem, showing that unbounded, almost periodic functions can be decomposed into unbounded periodic and bounded almost periodic parts.

Contribution

It introduces a simplified proof leveraging arithmetical properties of translation numbers, enhancing understanding of the theorem.

Findings

The proof confirms the decomposition of almost periodic functions into periodic and almost periodic components.

It demonstrates the applicability of arithmetical properties in function decomposition.

The approach simplifies previous proofs, making the theorem more accessible.

Abstract

We present a new elementary proof of a theorem due to Harald Bohr, which states that an unbounded, analytic, and almost periodic function in a half-plane can be written as the sum of two analytic functions: the first is unbounded and periodic, while the second is bounded and almost periodic. The proof is based on a well-known arithmetical property of translation numbers of almost periodic functions.