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Harald Bohr's splitting theorem

Viktor Andersson, Ole Fredrik Brevig, Athanasios Kouroupis

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.

Harald Bohr's splitting theorem

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.

Paper Structure

This paper contains 3 sections, 7 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Proofs
  3. Bohr's proof of the splitting theorem

Key Result

Theorem 1

The following are equivalent:

Theorems & Definitions (13)

  • Theorem 1: Bohr's splitting theorem
  • Theorem 2: Uniqueness in Bohr's splitting theorem
  • Lemma 3
  • Lemma 4
  • Theorem 5
  • proof : Proof of Lemma \ref{['lem:aritprog']}
  • proof : Proof of the implication (ii)$\implies$(i) in Theorem \ref{['thm:bohrsplit']}
  • Lemma 6
  • proof : Proof of Theorem \ref{['thm:percomp']}
  • Lemma 7
  • ...and 3 more