Towards Discrete Octonionic Analysis

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 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.

Plemelj projections in discrete quaternionic analysis

“…The landscape of octonions, marked by its non-associative nature, has yet to see a definitive blueprint for the discrete Cauchy integral and the discrete Plemelj projection operators. Krausshar and Legatiuk [24] took pioneering steps in this direction, a journey also echoed in [23].…”