Fueter's theorem in discrete Clifford analysis

Abstract

In this paper, we discretize techniques for the construction of axially monogenic functions to the setting of discrete Clifford analysis. Wherefore, we work in the discrete Hermitian Clifford setting, where each basis vector e j is split into a forward and backward basis vector: e j D e C j C e j . We prove a discrete version of Fueter's theorem in odd dimension by showing that for a discrete monogenic function f. 0 , 1 / left-monogenic in two variables 0 and 1 and for a left-monogenic P k . /, the m-dimensional function kC m 1 2 f. 0 , /P k . / is in itself left monogenic, that is, a discrete function in the kernel of the discrete Dirac operator. Closely related, we consider a Vekua-type system for the construction of axially monogenic functions. We consider some explicit examples: the discrete axial-exponential functions and the discrete Clifford-Hermite polynomials. Copyright © 2015 John Wiley & Sons, Ltd.Keywords: discrete Clifford analysis; axial monogenicity; Fueter theorem IntroductionIn 1935, the Swiss mathematician R. Fueter described in his paper This classical result was later generalized to R 0,m for m odd by Sce [2], for m even by Qian [3,4], and in [5], Sce's result was generalized by Sommen as follows: if m is an odd positive integer and P k .x/ is a homogeneous monogenic polynomial of degree k in R m , then is also monogenic in Q . The goal of this paper is to discretize Sommen's result to the setting of discrete Clifford analysis. Clifford analysis is a natural generalization of complex analysis to higher dimensions. For a detailed description, we refer to [6][7][8]. For the sake of completion, we recall some of the basic notions. It is constructed by associating to the standard Euclidean space R m the real Clifford algebra R 0,m , generated by the canonical basis e 1 , : : : , e m . These generators satisfy the multiplication rule e i e j C e j e i D 2 ı ij . For a basis of R 0,m , we consider for each set A D fj 1 , : : : , j k g  f1, : : : , mg the element e A D e j1 : : : e jk with 1 6 j 1 < j 2 < : : : < j k 6 m, together with e ; D 1, the identity element.The Euclidean space R m is embedded in R 0,m by identifying the point . Discrete Clifford analysis [9][10][11] has emerged in recent years as both a direct discretization of Euclidean Clifford analysis and a higherdimensional analogue of discrete complex analysis. There are, although, various choices related to the construction of a discrete Dirac operator. We work in the so-called discrete Hermitian setting, where the use of both forward and backward differences in the definition of the discrete Dirac operator @ translates into a direct factorization of the discrete (star) Laplacian: D @ 2 . As in Euclidean Clifford analysis, the construction of discrete monogenic functions, that is, functions in the kernel of the discrete Dirac operator, is a central topic in discrete Clifford analysis, and various techniques have already been developed or directly discretized: a discrete CauchyKovalevskaya extension theorem, a discrete Fischer decompo...