Direct sum Decomposition of Spaces of Periodic Functions: $$ \mathbb{P}_n = \bigoplus \limits_{d|n} \ker(\Phi_d(E))

Abstract

It was proved that the space $ \mathbb{P}_p $ of all periodic function of fundamental period $ p $ is a direct sum of the space $ \mathbb{P}_{p/2} $ of all periodic functions of fundamental period $ p/2 $ and the space $ \mathbb{AP}_{p/2} $ of all antiperiodic functions of fundamental antiperiod $ p/2 $. In this paper, we study some connections between periodic functions, cyclotomic polynomials, roots of unity, circulant matrices, and some classes of difference equations. In particular, we state and prove the sufficient condition for the existence of periodic solutions of integer period or arbitrary period of some difference equation. We also show that the space $ \mathbb{P}_n $ of all periodic functions of integer period $n$ can be decomposed as the direct sum of operators' kernels $\ker (\Phi_d(E)) $, where $\Phi_d(E),\, 1 \leq d \leq n, d|n $ are the cyclotomic polynomials of the shift operator $E$. We state and prove important theorems, state and prove the necessary and sufficient conditions for a linear difference equation with constant coefficients to have periodic solutions.