Motivated by questions about which functions could be represented by Dirichlet series, Harald Bohr founded the theory of almost periodic functions in the 1920s. This beautiful exposition begins with a discussion of periodic functions before addressing the almost periodic case. An appendix discusses almost periodic functions of a complex variable.

This is a beautiful exposition of the theory of Almost Periodic Functions written by the creator of that theory; translated by H. Cohn.

Table of Contents

• Introduction
• Purely periodic functions and their Fourier series: General orthogonal systems; Fourier constants with respect to a normal orthogonal system. Their minimal property. Bessel's formula and Bessel's inequality; Fourier series of periodic functions; Operations with Fourier series; Two fundamental theorems. The uniqueness theorem and Parseval's equation; Lebesgue's proof of the uniqueness theorem; The multiplication theorem; Summability of the Fourier series. Fejer's theorem; Weierstrass' theorem; Two remarks
• The theory of almost periodic functions: The main problem of the theory; Translation numbers; Definition of almost periodicity; Two simple properties of almost periodic functions; The invariance of almost periodicity under simple operations of calculation; The mean value theorem; The concept of the Fourier series of an almost periodic function. Derivation of Parseval's equation; Calculations with Fourier series; The uniqueness theorem. Its equivalence with Parseval's equation; The multiplication theorem; Introductory remarks to the proof of the two fundamental theorems; Preliminaries for the proof of the uniqueness theorem; Proof of the uniqueness theorem; The fundamental theorem; An important example
• Appendix I: Generalizations of almost periodic functions
• Appendix II: Almost periodic functions of a complex variable
• Bibliography