Harmonic Analysis and Hypercomplex Function Theory in Co-dimension One

• Main Author: Malonek, H.R.
• Publication Date: 2019
• Other Authors: Cação, I., Falcão, M.I., Tomaz, G.
• Format: Article
• Language: eng
• Source: Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)
• Download full: http://hdl.handle.net/10314/5210
• topic: Clifford algebras; Hypercomplex differential forms; Hypercomplex derivative; Hypercomplex Appell polynomials

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 their origins 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 are characterized 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 the other 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.