Application of the Weyl calculus perspective on discrete octonionic analysis in bounded domains

TL;DR

This paper develops the foundational aspects of discrete octonionic analysis in bounded domains using Weyl calculus, including formulas, Hardy spaces, and the role of non-associativity, establishing a basis for future research.

Contribution

It introduces a Weyl calculus-based framework for discrete octonionic analysis in bounded domains, including key formulas and Hardy spaces, completing the foundational development in this area.

Findings

Proved discrete Stokes formula for bounded cuboids.

Generalized formulas to arbitrary bounded domains.

Clarified the role of non-associativity in octonionic multiplication.

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.