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 paper, we establish a Cauchy-Kovalevskaya extension theorem for discrete monogenic functions defined on the standard $Z^m$ grid. Based on this extension principle, discrete Fueter polynomials, forming a basis of the space of discrete spherical monogenics, i.e. homogeneous discrete monogenic polynomials, are introduced. As an illustrative example we moreover explicitly construct the Cauchy-Kovalevskaya extension of the discrete delta function. These results are then generalized for a grid with variable mesh width $h$.