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On Polynomial Solutions of Generalized Moisil-Théodoresco Systems and Hodge-de Rham Systems

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Abstract

The aim of the paper is to study relations between polynomial solutions of generalized Moisil-Théodoresco (GMT) systems and polynomial solutions of Hodge-de Rham systems and, using these relations, to describe polynomial solutions of GMT systems. We decompose the space of homogeneous solutions of GMT system of a given homogeneity into irreducible pieces under the action of the group O(m) and we characterize individual pieces by their highest weights and we compute their dimensions.

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References

  1. R. Abreu Blaya, J. Bory Reyes, R. Delanghe and F. Sommen, Generalized Moisil- Théodoresco systems and Cauchy integral decompositions. Int. J. Math. Math. Sci., Vol. 2008, Article ID746946, 19 pages.

  2. Bock S., Gürlebeck K.: On a generalized Appell system and monogenic power series. Mathematical Methods in the Applied Sciences 33, 394–411 (2010)

    MathSciNet  MATH  Google Scholar 

  3. Brackx F., Delanghe R., Sommen F.: Differential forms and/or multi-vector functions. CUBO 7, 139–170 (2005)

    MathSciNet  MATH  Google Scholar 

  4. J. Bureš and V. Souček, Regular spinor valued mappings. Geometry seminars 1984 (Bologna, 1984), 7–23, Univ. Stud. Bologna, Bologna, 1985.

  5. I. Cação, Constructive approximation by monogenic polynomials. Ph.D-thesis, Univ. Aveiro, 2004.

  6. I. Cação and H. Malonek, On a complete set of hypercomplex Appell polynomials. Proc. ICNAAM 2008, (T. E. Timos, G. Psihoyios, Ch. Tsitouras, Eds.), AIP Conference Proceedings 1048, 647–650.

  7. Cialdea A.: The brothers Riesz theorem for conjugate differential forms in R n. Applicable Analysis 65, 69–94 (1997)

    Article  MathSciNet  Google Scholar 

  8. A. Cialdea, On the theory of self-conjugate differential forms. Atti Sem. Mat. Fis. Univ. Modena, Suppl. al Vol. XLVI (1998), 595–620.

  9. Delanghe R.: On homogeneous polynomial solutions of the Riesz system and their harmonic potentials. Complex Var. Elliptic Equ. 52(10-11), 1047–1061 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. R. Delanghe, On homogeneous polynomial solutions of generalized Moisil-Théodoresco systems in Euclidean space. preprint.

  11. Delanghe R., Sommen F., Souček V.: Clifford algebra and Spinor-Valued Functions. Kluwer, Dordrecht (1992)

    Book  MATH  Google Scholar 

  12. Fulton W., Harris J.: Representation theory. Springer, New York (1991)

    Book  MATH  Google Scholar 

  13. Gilbert J.E., Murray M.A.M.: Clifford Algebras and Dirac Operators in Harmonic Analysis. Cambridge University Press, Cambridge (1991)

    Book  MATH  Google Scholar 

  14. R. Goodman and N. R. Wallach, Representations and Invariants of the Classical Groups. Cambridge University Press, 1998.

  15. H. Leutwiler, Quaternionic analysis in R 3 versus its hyperbolic modification. In: F. Brackx, J. S. R. Chisholm and V. Souček (Eds.), Clifford analysis and its applications, Kluwer (2001), pp. 193–211.

  16. Homma Y.: Spinor-valued and Clifford algebra-valued harmonic polynomials. J. Geom. Phys. 37, 201–215 (2001)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. P. Zeitlinger, Beiträge zur Clifford Analysis und deren Modifikation. German, Ph.D-thesis, Univ. Erlangen, 2005.

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Authors and Affiliations

  1. Clifford Research Group, Department of Mathematical Analysis, Ghent University, Galglaan 2, B-9000, Gent, Belgium

    Richard Delanghe

  2. Mathematical Institute, Charles University, Sokolovská 83, 186 75, Praha 8, Czech Republic

    Roman Lávička & Vladimír Souček

Authors
  1. Richard Delanghe
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  2. Roman Lávička
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  3. Vladimír Souček
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Correspondence to Roman Lávička.

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Delanghe, R., Lávička, R. & Souček, V. On Polynomial Solutions of Generalized Moisil-Théodoresco Systems and Hodge-de Rham Systems. Adv. Appl. Clifford Algebras 21, 521–530 (2011). https://doi.org/10.1007/s00006-010-0262-4

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  • DOI: https://doi.org/10.1007/s00006-010-0262-4

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