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Keywords:
Fischer decomposition; Clifford analysis
Fischer decomposition; Clifford analysis
Summary:
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.
References:
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