The Cauchy-Kovalevskaya Extension Theorem in Discrete Clifford Analysis

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 Zh 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 CauchyKovalevskaya extensions of the discrete delta function and of a discretized exponential.