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In this paper we prove that the space $ \mathbb{P}_p $ of all periodicfunction of fundamental period $ p $ is a direct sum of the space $\mathbb{P}_{p/2} $ of periodic functions of fundamental period $ p/2 $ and thespace $ \mathbb{AP}_{p/2} $ of antiperiodic functions of fundamental antiperiod $ p/2 $. The decomposition can be continued by applying thedecomposition process to the successively raising periodic subspaces. It isshown that, under certain condition, a periodic function can be written as aconvergent infinite series of anti periodic functions of distinct fundamentalanti periods. In addition, we characterize the space of all periodic functionsof period $ p \in \mathbb{N} $ in terms of all its periodic and antiperiodicsubspaces of integer periods (or anti periods). We show that the elements of asubspace of such a space of periodic functions take a specific form (notarbitrary) of linear combinations of the shifts of the elements of the givenspace. Lastly, we introduce a lattice diagram named-periodicity diagram for aspace of periodic function of a fixed period $ p \in \mathbb{N} $. As aparticular example, the periodicity diagram of $ \mathbb{P}_{12} $ is shown.

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