Octonionic Dirac operators in bounded lattices: convergence and Plemlj projections

Abstract

This article explores octonionic analysis on the lattice $\mathbb Z^8$, emphasizing the octonionic discrete Cauchy integral within a bounded domain, the Sokhotski-Plemelj jump formulas, and the convergence of discrete regular functions. We address the challenge of articulating the discrete Cauchy-Pompeiu integral formula, particularly the complexities tied to the associator of octonions. Adopting an innovative approach, we assimilate complicating terms into integral kernels, leading to distinct surface and volume kernels. A new'star product' arises in handling octonionic function multiplications. Our study connects closely with the Teodorescu operator and singular kernel management, utilizing a Fourier variant with 24 unique singular points. Drawing from the foundational work of Shivakumar and Wong about the asymptotic expansion of the Fourier transform, we bridge the relationship between the Cauchy and Teodorescu integrals. This allows us to provide quantitative estimates for these kernels, which prove to be pivotal in the theory of discrete regular extensions. The research culminates in the revelation that a continuous octonionic function is regular precisely when it is a scale limit of discrete regular functions.