The Discrete Borel–Pompeiu Formula on a Rectangular Lattice

Abstract

Discrete function theory is a natural extension of the continuous theory to functions defined on lattices. The idea of the discrete function theory is to work directly with discretised domains (lattices) and to transfer all important properties from the continuous case to the discrete level. In the field of boundary value problems it is more beneficial to work with rectangular lattices, i.e. allowing two different stepsizes. Thus, the aim of this paper is to present the extension of the discrete function theory to rectangular lattices. Particularly, we present the discrete analogue of the Borel–Pompeiu formula. Analogously to the continuous theory it is the core of the developed discrete function theory and the related operator calculus.