Discrete Clifford analysis: an overview

Abstract

We give an account of our current research results in the development of a higher dimensional discrete function theory in a Clifford algebra context. On the simplest of all graphs, the rectangular ℤ^m grid, the concept of a discrete monogenic function is introduced. To this end new Clifford bases, involving so–called forward and backward basis vectors and introduced by means of their underlying metric, are controlling the support of the involved operators. As our discrete Dirac operator is seen to square up to a mixed discrete Laplacian, the resulting function theory may be interpreted as a refinement of discrete harmonic analysis. After a proper definition of some topological concepts, function theoretic results amongst which Cauchy‘s theorem and a Cauchy integral formula are obtained. Finally a first attempt is made at creating a general model for the Clifford bases used, involving geometrically interpretable curvature vectors.