Cauchy Formulae and Hardy Spaces in Discrete Octonionic Analysis

Abstract

In this paper, we continue the development of a fundament of discrete octonionic analysis that is associated to the discrete first order Cauchy–Riemann operator acting on octonions. In particular, we establish a discrete octonionic version of the Borel–Pompeiu formula and of Cauchy’s integral formula. The latter then is exploited to introduce a discrete monogenic octonionic Cauchy transform. This tool in hand allows us to introduce discrete octonionic Hardy spaces for upper and lower half-space together with Plemelj projection formulae.