Stone-Weierstrass Theorem

It will be shown that the Stone-Weierstrass theorem for Clifford-valued functions is true for the case of even dimension. It remains valid for the odd dimension if we add a stability condition by principal automorphism

The Legendre Formula in Clifford Analysis

2000 Mathematics Subject Classification: 30A05, 33E05, 30G30, 30G35, 33E20.Let R0,2m+1 be the Clifford algebra of the antieuclidean 2m+1 dimensional space. The elliptic Cliffordian functions may be generated by the z2m+2 function, analogous to the well-known Weierstrass z-function. The latter satisfies a Legendre equality. We prove a corresponding formula at the level of the monogenic function Dm z2m+2

Stone-Weierstrass theorem

It will be shown that the Stone-Weierstrass theorem for Clifford-valued functions is true for the case of even dimension. It remains valid for the odd dimension if we add a stability condition by principal automorphism

Jacobi Elliptic Cliffordian Functions

The well-known Jacobi elliptic functions sn(z)$,$cn(z), dn(z) are defined in higher dimensional spaces by the following method. Consider the Clifford algebra of the antieuclidean vector space of dimension 2m+1. Let x be the identity mapping on the space of scalars + vectors. The holomorphic Cliffordian functions may be viewed roughly as generated by the powers of x, namely x^n, their derivatives, their sums, their limits (cf : z^n for classical holomorphic functions). In that context it is possible to define the same type of functions as Jacobi's

Fonctions Holomorphes Cliffordiennes

Soit R_{0,2m+1} l'algèbre de Clifford de R^{2m+1} muni d'une forme quadratique de signature négative, D = \sum_{i=0}^{2m+1} e_i {\partial\over \partial x_i}, \Delta le Laplacien ordinaire. Les fonctions holomorphes Cliffordiennes f sont les fonctions satisfaisant à D\Delta^m f = 0. Nous étudions les solutions polynomiales et singulières, les représentations intégrales et leurs conséquences et enfin le fondement de la théorie des fonctions elliptiques Cliffordiennes

Holomorphic Cliffordian Functions

The aim of this paper is to put the fundations of a new theory of functions, called holomorphic Cliffordian, which should play an essential role in the generalization of holomorphic functions to higher dimensions. Let R_{0,2m+1} be the Clifford algebra of R^{2m+1} with a quadratic form of negative signature, D = \sum_{j=0}^{2m+1} e_j {\partial\over \partial x_j} be the usual operator for monogenic functions and $\Delta$ the ordinary Laplacian. The holomorphic Cliffordian functions are functions f : \R^{2m+2} \fle \R_{0,2m+1}, which are solutions of D \Delta^m f =

Elliptic Cliffordian Functions

In the study of holomorphic functions of one complex variable, one well-known theory is that of elliptic functions and it is possible to take the zeta-function of Weierstrass as a building stone of this vast theory. We are working the analogue theory in the natural context of higher dimensional spaces : holomorphic and elliptic Cliffordian functions