The Cauchy-Kovalevskaya Extension Theorem in Discrete Clifford Analysis

Abstract

Discrete Clifford analysis is a higher dimensional discrete function theory based on skew Weyl relations. It is centered around the study of Clifford algebra valued null solutions, called discrete monogenic functions, of a discrete Dirac operator, i.e. a first order, Clifford vector valued difference operator. In this contribution, we establish a Cauchy‐Kovalevskaya extension theorem for discrete monogenic functions defined on the grid Zhm of m‐tuples of integer multiples of a variable mesh width h. Convergence to the continuous case is investigated. As illustrative examples we explicitly construct the Cauchy‐Kovalevskaya extensions of the discrete delta function and of a discretized exponential.