Application of the Weyl calculus perspective on discrete octonionic analysis in bounded domains
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916384748863488 |
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| author | Kraußhar, Rolf Sören Legatiuk, Anastasiia Legatiuk, Dmitrii |
| author_facet | Kraußhar, Rolf Sören Legatiuk, Anastasiia Legatiuk, Dmitrii |
| contents | In this paper, we finish the basic development of the discrete octonionic analysis by presenting a Weyl calculus-based approach to bounded domains in $\mathbb{R}^{8}$. In particular, we explicitly prove the discrete Stokes formula for a bounded cuboid, and then we generalise this result to arbitrary bounded domains in interior and exterior settings by the help of characteristic functions. After that, discrete interior and exterior Borel-Pompeiu and Cauchy formulae are introduced. Finally, we recall the construction of discrete octonionic Hardy spaces for bounded domains. Moreover, we explicitly explain where the non-associativity of octonionic multiplication is essential and where it is not. Thus, this paper completes the basic framework of the discrete octonionic analysis introduced in previous papers, and, hence, provides a solid foundation for further studies in this field. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_04285 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Application of the Weyl calculus perspective on discrete octonionic analysis in bounded domains Kraußhar, Rolf Sören Legatiuk, Anastasiia Legatiuk, Dmitrii Analysis of PDEs Complex Variables 39A12, 42A38, 44A15 In this paper, we finish the basic development of the discrete octonionic analysis by presenting a Weyl calculus-based approach to bounded domains in $\mathbb{R}^{8}$. In particular, we explicitly prove the discrete Stokes formula for a bounded cuboid, and then we generalise this result to arbitrary bounded domains in interior and exterior settings by the help of characteristic functions. After that, discrete interior and exterior Borel-Pompeiu and Cauchy formulae are introduced. Finally, we recall the construction of discrete octonionic Hardy spaces for bounded domains. Moreover, we explicitly explain where the non-associativity of octonionic multiplication is essential and where it is not. Thus, this paper completes the basic framework of the discrete octonionic analysis introduced in previous papers, and, hence, provides a solid foundation for further studies in this field. |
| title | Application of the Weyl calculus perspective on discrete octonionic analysis in bounded domains |
| topic | Analysis of PDEs Complex Variables 39A12, 42A38, 44A15 |
| url | https://arxiv.org/abs/2409.04285 |