Fischer decompositions in Euclidean and Hermitean Clifford analysis

Fred Brackx * , Hennie De Schepper * , and Vladimír Souček ‡

* Clifford Research Group, Faculty of Engineering, Ghent University, Galglaan 2, B-9000 Gent, Belgium
‡ Mathematical Institute, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha, Czech Republic


Abstract: Euclidean Clifford analysis is a higher dimensional function theory studying so–called monogenic functions, i.e. null solutions of the rotation invariant, vector valued, first order Dirac operator $\underline{\partial }$. In the more recent branch Hermitean Clifford analysis, this rotational invariance has been broken by introducing a complex structure $J$ on Euclidean space and a corresponding second Dirac operator $\underline{\partial }_J$, leading to the system of equations $\underline{\partial } f = 0 = \underline{\partial }_J f$ expressing so-called Hermitean monogenicity. The invariance of this system is reduced to the unitary group U($n$). In this paper we decompose the spaces of homogeneous monogenic polynomials into U($n$)-irrucibles involving homogeneous Hermitean monogenic polynomials and we carry out a dimensional analysis of those spaces. Meanwhile an overview is given of so-called Fischer decompositions in Euclidean and Hermitean Clifford analysis.

AMSclassification: primary 30G35.

Keywords: Fischer decomposition, Clifford analysis.