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This paper is a short report on the generalization of some results of our previous paper (12) to the case of spin j /2 Dirac operators in real dimension three for arbitrary odd integer j . We use an explicit formula for the local expression of such operators to study their algebraic properties,...
Following the previous study on the unit ball of Delanghe et al, half-Dirichlet problems for the upper-half space are presented and solved. The solutions further lead to decompositions of the Poisson kernels, and the fact that the classical Dirichlet problems may be solved merely by using Cauchy...
The paper deals with sampling of σ-bandlimited functions in R m with Clifford-valued, where bandlimitedness means that the spectrum is contained in the ball B (0, σ ) that is centered at the origin and has radius σ . By comparing with the general setting, what is new in the sampling is using...
Let F r (0 < r < m + 1) be a smooth r -vector valued function in a suitable open domain of satisfying in Ω, where ∂ is the Dirac operator in . Then it is proved that there exists H r , an r -vector valued harmonic function in Ω, such that F r = . Two proofs of this structure theorem are...
The theory of functions with values in the algebra of quaternions shows a lot of analogies to the function theory in the complex one-dimensional case. The class of holomorphic functions is replaced by the set of null solutions of a generalized Cauchy-Riemann system, the class of monogenic...
In this paper, we show the existence of a sequence of invariant differential operators on a particular homogeneous model G / P of a Cartan geometry. The first operator in this sequence is locally the Dirac operator in 2 Clifford variables, D = ( D 1 , D 2 ), where D i = ∑ j e j . ∂ ij . It...
The aim of this article is to consider the hyperbolic version of the standard Clifford analysis. The need for such a modification arises when one wants to make sure that the power function x m is included. The leading idea is that the power function is the conjugate gradient of a harmonic...
We treat the quaternionic Fourier transform (QFT) applied to quaternion fields and investigate QFT properties useful for applications. Different forms of the QFT lead us to different Plancherel theorems. We relate the QFT computation for quaternion fields to the QFT of real signals. We research...
Orthogonal Clifford analysis in flat m –dimensional Euclidean space focusses on monogenic functions, i.e. null solutions of the rotation invariant vector valued Dirac operator , where ( ) forms an orthogonal basis for the quadratic space underlying the construction of the Clifford algebra ....
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