{"title":"Application of the Weyl calculus perspective on discrete octonionic analysis in bounded domains","authors":"Rolf Sören Kraußhar, Anastasiia Legatiuk, Dmitrii Legatiuk","doi":"arxiv-2409.04285","DOIUrl":null,"url":null,"abstract":"In this paper, we finish the basic development of the discrete octonionic\nanalysis by presenting a Weyl calculus-based approach to bounded domains in\n$\\mathbb{R}^{8}$. In particular, we explicitly prove the discrete Stokes\nformula for a bounded cuboid, and then we generalise this result to arbitrary\nbounded domains in interior and exterior settings by the help of characteristic\nfunctions. After that, discrete interior and exterior Borel-Pompeiu and Cauchy\nformulae are introduced. Finally, we recall the construction of discrete\noctonionic Hardy spaces for bounded domains. Moreover, we explicitly explain\nwhere the non-associativity of octonionic multiplication is essential and where\nit is not. Thus, this paper completes the basic framework of the discrete\noctonionic analysis introduced in previous papers, and, hence, provides a solid\nfoundation for further studies in this field.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"15 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Complex Variables","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.04285","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
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.