Skip to main content
SpringerLink
Log in

On representations of real analytic functions by monogenic functions

  • Published:
Czechoslovak Mathematical Journal Aims and scope Submit manuscript

Abstract

Using the method of normalized systems of functions, we study one representation of real analytic functions by monogenic functions (i.e., solutions of Dirac equations), which is an Almansi’s formula of infinite order. As applications of the representation, we construct solutions of the inhomogeneous Dirac and poly-Dirac equations in Clifford analysis.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Z. R. Al-Yasiri, K. Gůrlebeck: On a boundary value problem for a p-Dirac equation. Math. Methods Appl. Sci. 39 (2016), 4056–4068.

    Article  MathSciNet  Google Scholar 

  2. N. Aronszajn, T. M. Creese, L. J. Lipkin: Polyharmonic Functions. Oxford Mathematical Monographs, Oxford University Press, Oxford, 1983.

    Google Scholar 

  3. B. A. Bondarenko: Operator Algorithms in Differential Equations. Izdatel’stvo Fan Uzbekskoj SSR, Tashkent, 1984. (In Russian.)

    MATH  Google Scholar 

  4. F. Brackx, R. Delanghe, F. Sommen: Clifford Analysis. Research Notes in Mathematics 76, Pitman Advanced Publishing Program, Boston, 1982.

    Google Scholar 

  5. F. Brackx, H. De Schepper, D. Eelbode, V. Souček: Explicit formulae for monogenic projections. Int. Conf. on Numerical Analysis and Applied Mathematics 2008 (T. Simos et al., eds.). AIP Conference Proceedings 1048, American Institute of Physics, Melville, 2008, pp. 697–700.

    Google Scholar 

  6. D. Constales, D. Grob, R. S. Kraußhar: Reproducing kernel functions of solutions to polynomial Dirac equations in the annulus of the unit ball in ℝn and applications to boundary value problems. J. Math. Anal. Appl. 358 (2009), 281–293.

    Article  MathSciNet  Google Scholar 

  7. R. Delanghe, F. Sommen, V. Souček: Clifford Algebra and Spinor-Valued Functions. A Function Theory for the Dirac Operator. Mathematics and Its Applications 53, Kluwer Academic Publishers, Dordrecht, 1992

    MATH  Google Scholar 

  8. R. Howe: Remarks on classical invariant theory. Trans. Am. Math. Soc. 313 (1989), 539–570.

    Article  MathSciNet  Google Scholar 

  9. U. Kähler: Clifford analysis and the Navier-Stokes equations over unbounded domains. Adv. Appl. Clifford Algebr. 11 (2001), 305–318.

    Article  MathSciNet  Google Scholar 

  10. V. V. Karachik: Polynomial solutions to systems of partial differential equations with constant coefficients. Yokohama Math. J. 47 (2000), 121–142.

    MathSciNet  MATH  Google Scholar 

  11. V. V. Karachik: Normalized system of functions with respect to the Laplace operator and its applications. J. Math. Anal. Appl. 287 (2003), 577–592.

    Article  MathSciNet  Google Scholar 

  12. V. V. Karachik: Method of Normalized Systems of Functions. Izd. Tsentr Yuzhno-Ural’skiľ Gosudarstvennyľ Universitet, Chelyabinsk, 2014. (In Russian.)

    MATH  Google Scholar 

  13. V. V. Karachik: Solution of the Dirichlet problem with polynomial data for the polyharmonic equation in a ball. Differ. Equ. 51 (2015), 1033–1042 (In English. Russian original.); translation from Differ. Uravn. 51 (2015), 1038–1047.

    Article  MathSciNet  Google Scholar 

  14. V. V. Karachik, B. Turmetov: Solvability of some Neumann-type boundary value problems for biharmonic equations. Electron. J. Differ. Equ. 217 (2017), Paper No. 218, 17 pages.

  15. M. Ku, D. Wang: Solutions to the polynomial Dirac equations on unbounded domains in Clifford analysis. Math. Methods Appl. Sci. 34 (2011), 418–427.

    MathSciNet  MATH  Google Scholar 

  16. J. Ryan: Cauchy-Green type formulae in Clifford analysis. Trans. Am. Math. Soc. 347 (1995), 1331–1341.

    Article  MathSciNet  Google Scholar 

  17. F. Sommen, B. Jancewicz: Explicit solutions of the inhomogeneous Dirac equation. J. Anal. Math. 71 (1997), 59–74.

    Article  MathSciNet  Google Scholar 

  18. H. F. Yuan: Dirichlet type problems for Dunkl-Poisson equations. Bound. Value Probl. 2016 (2016), Article ID 222, 16 pages.

    Google Scholar 

  19. H. F. Yuan: Solutions of the Poisson equation and related equations in super spinor space. Comput. Methods Funct. Theory 16 (2016), 699–715.

    Article  MathSciNet  Google Scholar 

  20. H. F. Yuan, V. V. Karachik: Dunkl-Poisson equation and related equations in superspace. Math. Model. Anal. 20 (2015), 768–781.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author would like to thank Valery V. Karachik for helpful discussions on the method of normalized systems of functions.

Author information

Authors and Affiliations

  1. School of Mathematics and Physics, Hebei University of Engineering, Guangming South Street 199, Handan, Hebei Prov., 056038, P. R. China

    Hongfen Yuan

Authors
  1. Hongfen Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hongfen Yuan.

Additional information

This work was supported by the NNSF of China (No. 11426082), the Natural Science Foundation of Hebei Province (No. A2016402034), Project of Handan Municipal Science and Technology Bureau (No. 1534201097-10), and Foundation of Hebei University of Engineering under Grant Nos. 16121002014, 17129033049, 86210022, 00070348.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yuan, H. On representations of real analytic functions by monogenic functions. Czech Math J 69, 997–1013 (2019). https://doi.org/10.21136/CMJ.2019.0573-17

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.21136/CMJ.2019.0573-17

Keywords

MSC 2010

Search

Navigation