Decomposition of Spaces of Periodic Functions into Subspaces of Periodic Functions and Subspaces of Antiperiodic Functions
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909687357636608 |
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| author | Yadeta, Hailu Bikila |
| author_facet | Yadeta, Hailu Bikila |
| contents | 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} $. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_00915 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Decomposition of Spaces of Periodic Functions into Subspaces of Periodic Functions and Subspaces of Antiperiodic Functions Yadeta, Hailu Bikila General Mathematics 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} $. |
| title | Decomposition of Spaces of Periodic Functions into Subspaces of Periodic Functions and Subspaces of Antiperiodic Functions |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2210.00915 |