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Duality for Hermitean Systems in \({\mathbb R^{2n}}\)

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Abstract

In this paper, using the ring structure of the space of circulant (2 × 2)-matrix, we characterize the dual of the (Fréchet) space of germs of left Hermitean monogenic matrix functions in a compact set \({{\bf E}\subset\mathbb R^{2n}}\). As an application we describe the dual space of the so-called h-monogenic functions satisfying simultaneously two Dirac type equations.

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References

  1. Abreu Blaya, R., Bory Reyes, J., Brackx, F., De Knock, B.H., De Schepper, H., Peña Peña, D., Sommen, F.: Hermitean Cauchy integral decomposition of continuous functions on hypersurfaces. Bound. Value Probl. vol. 2008, 16 pp, Article ID 425256 (2008)

  2. Abreu Blaya, R., Bory Reyes, J., Brackx, F., De Schepper, H.: Hermitean Théodoresco transform decomposition of continuous matrix functions on fractal hypersurfaces. Bound. Value Probl. vol. 2010, 15 pp, Article ID 791358 (2010)

  3. Abreu Blaya, R., Bory Reyes, J., Peña Peña, D., Sommen, F.: A boundary value problem for Hermitian monogenic functions. Bound. Value Probl., vol. 2008, 7 pp, Article ID 385874 (2008)

  4. Abreu Blaya R., Bory Reyes J., Delanghe R., Sommen F.: Duality for harmonic differential forms via Clifford analysis. Adv. Appl. Clifford Algebr. 17(4), 589–610 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Abreu Blaya R., Bory Reyes J., Moreno García T.: Hermitian decomposition of continuous functions on a fractal surface. Bull. Braz. Math. Soc. 40(1), 107–115 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Delanghe R., Brackx F.: Duality in hypercomplex function theory. J. Funct. Anal. 37(2), 164–181 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brackx F., Delanghe R., Sommen F.: Clifford Analysis. Research Notes in Mathematics, vol. 76. Pitman, Boston (1982)

    Google Scholar 

  8. Brackx F., Bureš J., De Schepper H., Eelbode D., Sommen F., Souček V.: Fundaments of Hermitean Clifford analysis. II. Splitting of h-monogenic equations. Complex Var. Elliptic Equ. 52(10-11), 1063–1079 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Brackx F., Bureš J., De Schepper H., Eelbode D., Sommen F., Souček V.: Fundaments of Hermitean Clifford analysis. I. Complex structure. Complex Anal. Oper. Theory 1(3), 341–365 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Brackx F., De Schepper H., De Schepper N., Sommen F.: Hermitian Clifford-Hermite wavelets: an alternative approach. Bull. Belg. Math. Soc. Simon Stevin 15(1), 87–107 (2008)

    MathSciNet  MATH  Google Scholar 

  11. Brackx F., De Schepper Hennie, Sommen F.: The Hermitian Clifford analysis toolbox. Adv. Appl. Clifford Algebr. 18(3-4), 451–487 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Brackx, F., De Schepper, H., Luna-Elizarraras, M.E., Shapiro, M.: Integral representation formulae in Hermitean Clifford analysis. In: Gürlebeck, K., Könke, C. (eds.) 18th International Conference on the Application of Computer Science and Mathematics in Architecture and Civil Engineering. Weimar, Germany (07–09 July 2009)

  13. Brackx F., De Knock B., De Schepper H.: A matrix Hilbert transform in Hermitean Clifford analysis. J. Math. Anal. Appl. 344, 1068–1078 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Brackx F., De Knock B., De Schepper H., Sommen F.: On Cauchy and Martinelli–Bochner integral formulae in Hermitean Clifford analysis. Bull. Braz. Math. Soc. 40(3), 395–416 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Colombo F., Sabadini I., Sommen F., Struppa D.: Analysis of Dirac Systems and Computational Algebra. Birkhauser, Boston (2004)

    Book  MATH  Google Scholar 

  16. Dáger R., Presa A.: Duality of the space of germs of harmonic vector fields on a compact. C. R. Math. Acad. Sci. Paris 343(1), 19–22 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Dáger, R., Presa, A.: Duality of the space of germs of harmonic vector fields. arXiv: math. FA/0610924v1, 30 Octocber (2006)

  18. Gustafsson B., Khavinson D.: On annihilators of harmonic vector fields. Zap. Nauchn. Sem. POMI 232, 90–108 (1996)

    Google Scholar 

  19. Herrera Torres A., Presa A.: Approximation properties of holomorphic pairs of differential forms. Complex Var. Theory Appl. 50(2), 89–1001 (2005)

    MathSciNet  MATH  Google Scholar 

  20. Köthe G.: Dualität in der Funktionentheorie. J. Reine Angew. Math. 191, 30–49 (1953)

    MathSciNet  MATH  Google Scholar 

  21. Fabiano D.: Boundary values of monogenic functions. Adv. Appl. Clifford Algebr. 6(2), 193–206 (1996)

    MathSciNet  MATH  Google Scholar 

  22. Rocha-Chávez, R., Shapiro, M., Sommen, F.: Integral theorems for functions and differential forms in \({\mathbb C^m}\). Chapman & Hall/CRC Research Notes in Mathematics, pp. 428. Chpaman & Hall/CRC, Boca Raton (2002) (x+204 pp. ISBN: 1-58488-246-8)

  23. Ryan J.: Duality in complex Clifford analysis. J. Funct. Anal. 61, 117–135 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  24. Sabadini I., Sommen F.: Hermitian Clifford analysis and resolutions. Math. Methods Appl. Sci. 25(16–18), 1395–1413 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sommen F.: Hyperfunctions with values in a Clifford algebra. Simon Stevin 57, 225–254 (1983)

    MathSciNet  MATH  Google Scholar 

  26. Sommen F.: Microfunctions with values in a Clifford algebra II. Sci. Pap. College Arts Sci. Univ. Tokyo 36, 15–37 (1986)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Departamento de Matemática, Universidad de Holguín, Holguín, 80100, Cuba

    Ricardo Abreu-Blaya

  2. Departamento de Matemática, Universidad de Oriente, Santiago de Cuba, 90500, Cuba

    Juan Bory-Reyes

  3. Department of Mathematical Analysis, Faculty of Engineering, Ghent University, Galglaan 2, 9000, Gent, Belgium

    Richard Delanghe & Frank Sommen

Authors
  1. Ricardo Abreu-Blaya
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  2. Juan Bory-Reyes
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  3. Richard Delanghe
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  4. Frank Sommen
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Corresponding author

Correspondence to Ricardo Abreu-Blaya.

Additional information

Communicated by Daniel Aron Alpay.

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Abreu-Blaya, R., Bory-Reyes, J., Delanghe, R. et al. Duality for Hermitean Systems in \({\mathbb R^{2n}}\) . Complex Anal. Oper. Theory 6, 341–357 (2012). https://doi.org/10.1007/s11785-010-0098-x

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  • DOI: https://doi.org/10.1007/s11785-010-0098-x

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