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

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 ${\mathbb{P}}_{p\mathrm{/}2}$ of all periodic functions with fundamental period $p\mathrm{/}2$ and the space ${\mathbb{A}\mathbb{P}}_{p\mathrm{/}2}$ of all antiperiodic functions with fundamental antiperiod $p\mathrm{/}2$. 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 ${\mathbb{P}}_{12}$.