Harmonic Analysis and hypercomplex function theory in co-dimension one

Abstract

Fundamentals of a function theory in co-dimension one for Clifford algebra valued functions over R^(n+1) are considered. Special attention is given to theirorigins in analytic properties of holomorphic functions of one and, by some duality reasons, also of several complex variables. Due to algebraic peculiarities caused by non-commutativity of the Clifford product, generalized holomorphic functions arecharacterized by two different but equivalent properties: on one side by local derivability (existence of a well defined derivative related to co-dimension one) and on theother side by differentiability (existence of a local approximation by linear mappings related to dimension one). As important applications, sequences of harmonic Appell polynomials are considered whose definition and explicit analytic representations rely essentially on both dual approaches.