English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

It was proved that the space $ \mathbb{P}_p $ of all periodic function offundamental period $ p $ is a direct sum of the space $ \mathbb{P}_{p/2} $ ofall 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 ofdifference equations. In particular, we state and prove the sufficientcondition for the existence of periodic solutions of integer period orarbitrary period of some difference equation. We also show that the space $\mathbb{P}_n $ of all periodic functions of integer period $n$ can bedecomposed 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 theshift operator $E$. We state and prove important theorems, state and prove thenecessary and sufficient conditions for a linear difference equation withconstant coefficients to have periodic solutions.

Direct sum decomposition of spaces of periodic functions and some  connections between shift operators, periodicity of solutions of difference  equations, circulant matrices, cyclotomic polynomials, and roots of unity