Direct sum Decomposition of Spaces of Periodic Functions: ℙ_n = \bigoplus _d|n ker(Φ_d(E))

2023/04/05 by Hailu Bikila Yadeta, Yadeta, Hailu Bikila
Mathematics · #11C20 #11R60 #39A06 #39A23 #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2304.02517

openalex publication_date 2023/04/05 · openalex created_date 2023/04/07 · openalex updated_date 2026/07/28

Abstract

It was proved that the space ℙ_p of all periodic function of fundamental period p is a direct sum of the space ℙ_p/2 of all periodic functions of fundamental period p/2 and the space \mathbbAP_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 ℙ_n of all periodic functions of integer period n can be decomposed as the direct sum of operators' kernels ker (Φ_d(E)) , where Φ_d(E), 1 ≤ d ≤ 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.

Related

• The periodic decomposition problem
• On the difference equations with periodic coefficients
• On periodic approximate solutions of ordinary differential equations
• On sums and products of periodic functions
• Decomposition of Spaces of Periodic Functions into Subspaces of Periodic and Antiperiodic Functions and Its Connection to the Rademacher System and the Haar Wavelet Basis