Fischer decompositions in Euclidean and Hermitean Clifford analysis

Freddy Brackx; Hennie de Schepper; Vladimír Souček

Archivum Mathematicum (2010)

  • Volume: 046, Issue: 5, page 301-321
  • ISSN: 0044-8753

Abstract

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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 ∂ ̲ . 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 ∂ ̲ J , leading to the system of equations ∂ ̲ f = 0 = ∂ ̲ 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.

How to cite

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Brackx, Freddy, de Schepper, Hennie, and Souček, Vladimír. "Fischer decompositions in Euclidean and Hermitean Clifford analysis." Archivum Mathematicum 046.5 (2010): 301-321. <http://eudml.org/doc/116494>.

@article{Brackx2010,
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.},
author = {Brackx, Freddy, de Schepper, Hennie, Souček, Vladimír},
journal = {Archivum Mathematicum},
keywords = {Fischer decomposition; Clifford analysis; Fischer decomposition; Clifford analysis},
language = {eng},
number = {5},
pages = {301-321},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Fischer decompositions in Euclidean and Hermitean Clifford analysis},
url = {http://eudml.org/doc/116494},
volume = {046},
year = {2010},
}

TY - JOUR
AU - Brackx, Freddy
AU - de Schepper, Hennie
AU - Souček, Vladimír
TI - Fischer decompositions in Euclidean and Hermitean Clifford analysis
JO - Archivum Mathematicum
PY - 2010
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 046
IS - 5
SP - 301
EP - 321
AB - 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.
LA - eng
KW - Fischer decomposition; Clifford analysis; Fischer decomposition; Clifford analysis
UR - http://eudml.org/doc/116494
ER -

References

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