Decomposition of Spaces of Periodic Functions into Subspaces of Periodic Functions and Subspaces of Antiperiodic Functions

TL;DR

This paper explores how spaces of periodic functions can be decomposed into subspaces of smaller periodic and antiperiodic functions, providing a detailed structure and visualization tools for understanding these relationships.

Contribution

It introduces a novel decomposition of periodic function spaces into periodic and antiperiodic subspaces, with an iterative process and a new visualization called the periodicity diagram.

Findings

Decomposition of periodic spaces into subspaces of smaller periods and antiperiods.

Representation of periodic functions as series of antiperiodic functions.

Introduction of the periodicity diagram for visualizing relationships.

Abstract

In this paper, we establish that the space $ \mathbb{P}_p $ of all periodic function of fundamental period $ p $ can be expressed as a direct sum of the space $ \mathbb{P}_{p/2} $ of all periodic functions with fundamental period $ p/2 $ and the space $ \mathbb{AP}_{p/2} $ of all antiperiodic functions with fundamental antiperiod $ p/2 $. This decomposition process can be iteratively applied to successively refined periodic subspaces. We demonstrate that, under certain conditions, any periodic function can be represented as a convergent infinite series of antiperiodic functions with distinct fundamental antiperiods. Furthermore, we characterize the space of all periodic functions with period $ p \in \mathbb{N} $ in terms of its periodic and antiperiodic subspaces associated with integer periods (or antiperiods). We show that elements belonging to a subspace of such a space assume a specific structure: linear combinations of shifted versions of the basis functions, rather than arbitrary combinations. Finally, we introduce a lattice diagram called \emph{\emph{periodicity diagram}} to visualize the relationships within a space of periodic functions with a fixed period $ p \in \mathbb{N} $. As an illustrative example, we present the periodicity diagram for $ \mathbb{P}_{12} $.