论文标题
周期函数空间的分解为周期函数的子空间和反碘函数的子空间
Decomposition of Spaces of Periodic Functions into Subspaces of Periodic Functions and Subspaces of Antiperiodic Functions
论文作者
论文摘要
在本文中，我们确定所有基本期间$ p $的所有定期功能的空间可以表示为所有周期函数的直接总和$ \ mathbb {p} _ {p/2} $，所有周期性函数均具有基本的$ p/2 $ p/2 $ p/2 $ p/p/mathbbbbbbbbbbbbbbbbbbbbbb {p/具有基本反eod $ p/2 $的功能。该分解过程可以迭代地应用于连续完善的周期子空间。我们证明，在某些条件下，任何周期性的功能都可以表示为具有不同基本抗牙本质的一系列无限无限的反碘功能。 此外，我们以\ mathbb {n} $ in \ in \ mathbb {n} $的周期性和抗抑制性子空间（或anteger时期（或antiperiods）相关的周期性和反静脉子空间）来表征所有周期函数的空间。我们表明，属于这样一个空间的子空间的元素假设一个特定的结构：基本函数的移位版本的线性组合，而不是任意组合。 最后，我们介绍了一个晶格图，称为\ emph {\ emph {presceicity图}}，以可视化具有固定周期$ p \ in \ mathbb {n} $的周期函数空间内的关系。作为说明性示例，我们介绍了$ \ mathbb {p} _ {12} $的周期性图。
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/2} $ of all periodic functions with fundamental period $ p/2 $ and the space $ \mathbb{AP}_{p/2} $ of all antiperiodic functions with fundamental antiperiod $ p/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} $.
