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On a Mixed Fischer Decomposition in Clifford Analysis

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Abstract

In this paper, we investigate the existence of a mixed Fischer decomposition in a generalized Clifford analysis setting, where a system of Dirac equations is considered, associated to different orthogonal bases (or structural sets) in Euclidean space.

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References

  1. Abreu-Blaya, R., Bory-Reyes, J., Guzmán-Adán, A., Kaehler, U.: On some structural sets and a quaternionic (\(\varphi,\psi \))-hyperholomorphic function theory. Math. Nachr. 288(13), 1451–1475 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Axler, S., Gorkin, P., Voss, K.: The Dirichlet problem on quadratic surfaces. Math. Comp. 73(246), 637–651 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baker, J.A.: The Dirichlet problem for ellipsoids. Am. Math. Monthly 106, 829–834 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bory-Reyes, J., De Schepper, H., Guzmán-Adán, A., Sommen, F.: Higher order Borel-Pompeiu representations in Clifford Analysis. Math. Meth. Appl. Sci. (2015). doi:10.1002/mma.3798

  5. Brackx, F.: Teorie van de (k)-monogene funkties van een kwaternionenveranderlijke, PhD dissertation (in Dutch), Ghent University, (1973)

  6. Brackx, F.: On (k)-monogenic functions of a quaternion variable. In: Gilbert, R.P., Weinacht, R.J. (eds.) Function theoretic methods in differential equations, pp. 22–44. Pitman Publ., London (1976)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Brackx, F., Delanghe, R., Sommen, F.: Clifford analysis, research notes in mathematics, vol. 76. Pitman (Advanced Publishing Program), Boston (1982)

    MATH  Google Scholar 

  9. Brackx, F., De Schepper, H., Sommen, F.: The Hermitian Clifford analysis toolbox. Adv. Appl. Clifford Algebras 18(3–4), 451–487 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fischer, E.: Uber die Differentiationsprozesse der Algebra. J. für Math. 148, 1–78 (1917)

    MATH  Google Scholar 

  11. Gürlebeck, K., Habetha, K., Sprössig, W.: Holomorphic functions in the plane and n-dimensional space. Birkhäuser Verlag, Basel, Boston, Berlin (2008)

    MATH  Google Scholar 

  12. Khavinson, D., Shapiro, H.S.: Dirichlet’s problem when the data is an entire function. Bull. Lond. Math. Soc. 24, 456–468 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  13. Shapiro, H.S.: An algebraic theorem of E. Fischer, and the holomorphic Goursat problem. Bull. Lond. Math. Soc. 21, 513–537 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  14. Shapiro, M.V.: On some boundary value problems for functions with values in Clifford algebras. Matem. Vesnik. Beograd 40, 321–326 (1988)

    MathSciNet  MATH  Google Scholar 

  15. Shapiro, M.V.: On the conjugate harmonic functions of M. Riesz-E. Stein-G. Weiss. In: Dimiev, S., et al. (eds.), Topics in complex analysis, differential geometry and mathematical physics (3rd international workshop on complex structures and vector fields), World Scientific, pp. 8–32 (1997)

  16. Sudbery, A.: Quaternionic analysis. Math. Proc. Camb. Philos. Soc. 85, 199–225 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  17. Vasilevsky, N.L., Shapiro, M.V.: Some questions of hypercomplex analysis. Complex Analysis and Applications ’87 (Varna, 1987), Publ. House Bulgar. Acad. Sci., Sofia 1989, pp. 523–531 (1987)

  18. Vasilevski, N.L., Shapiro, M.V.: Holomorphy, hyperholomorphy, Teoplitz operators. Russ. Math. Surv. 44(4), 196–197 (1989)

    Google Scholar 

  19. Shapiro, M.V., Vasilevski, N.L.: Quaternionic \(\psi \)-hyperholomorphic functions, singular integral operators and boundary value problems I. \(\psi \)-hyperholomorphic function theory. Complex Var. Theory Appl. 27, 17–46 (1995)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. ESIME-Zacatenco, Instituto Politécnico Nacional, México, DF, 07738, Mexico

    Juan Bory Reyes

  2. Clifford Research Group, Department of Mathematical Analysis, Faculty of Engineering and Architecture, Ghent University, Galglaan 2, 9000, Ghent, Belgium

    Hennie De Schepper, Alí Guzmán Adán & Frank Sommen

Authors
  1. Juan Bory Reyes
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  2. Hennie De Schepper
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  3. Alí Guzmán Adán
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  4. Frank Sommen
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Corresponding author

Correspondence to Alí Guzmán Adán.

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Communicated by Irene Sabadini.

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Reyes, J.B., De Schepper, H., Adán, A.G. et al. On a Mixed Fischer Decomposition in Clifford Analysis. Complex Anal. Oper. Theory 11, 359–374 (2017). https://doi.org/10.1007/s11785-016-0574-z

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  • DOI: https://doi.org/10.1007/s11785-016-0574-z

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