Advertisement

Complex Analysis and Operator Theory

, Volume 6, Issue 2, pp 477–489 | Cite as

Complete Orthogonal Appell Systems for Spherical Monogenics

  • R. LávičkaEmail author
Article

Abstract

In this paper, we investigate properties of Gelfand–Tsetlin bases mainly for spherical monogenics, that is, for spinor valued or Clifford algebra valued homogeneous solutions of the Dirac equation in the Euclidean space. Recently it has been observed that in dimension 3 these bases form an Appell system. We show that Gelfand–Tsetlin bases of spherical monogenics form complete orthogonal Appell systems in any dimension. Moreover, we study the corresponding Taylor series expansions for monogenic functions. We obtain analogous results for spherical harmonics as well.

Keywords

Spherical harmonics Spherical monogenics Gelfand–Tsetlin basis Appell system Orthogonal basis Taylor series 

Mathematics Subject Classification (2000)

30G35 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Andrews G.E., Askey R., Roy R.: Special Functions, Encyclopedia of Mathematics and its Applications, vol. 71. Cambridge University Press, Cambridge (1999)Google Scholar
  2. 2.
    Bock S.: Orthogonal Appell bases in dimension 2,3 and 4. In: Simos, T.E., Psihoyios, G., Tsitouras, Ch. (eds) Numerical Analysis and Applied Mathematics, AIP Conference Proceedings, vol. 1281, pp. 1447–1450. American Institute of Physics, Melville, NY (2010)Google Scholar
  3. 3.
    Bock, S.: Über funktionentheoretische Methoden in der räumlichen Elastizitätstheorie (German), Ph.D thesis, Bauhaus-University, Weimar. http://e-pub.uni-weimar.de/frontdoor.php?source_opus=1503 (2009)
  4. 4.
    Bock S., Gürlebeck K.: On a generalized Appell system and monogenic power series. Math. Methods Appl. Sci. 33, 394–411 (2010)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bock, S., Gürlebeck, K., Lávička, R., Souček, V.: The Gel’fand-Tsetlin bases for spherical monogenics in dimension 3 (to appear in Rev. Mat. Iberoamericana, 2010). arXiv:1010.1615v2 [math.CV]Google Scholar
  6. 6.
    Brackx F., Delanghe R., Sommen F.: Clifford analysis. Pitman, London (1982)zbMATHGoogle Scholar
  7. 7.
    Brackx F., De Schepper H., Lávička R., Souček V.: The Cauchy–Kovalevskaya Extension Theorem in Hermitean Clifford Analysis. J. Math. Anal. Appl. 381, 649–660 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Brackx, F., De Schepper, H., Lávička, R., Souček, V.: Gelfand–Tsetlin Bases of Orthogonal Polynomials in Hermitean Clifford Analysis (to appear in Math. Methods Appl. Sci., 2011). arXiv:1102.4211v1 [math.CV]Google Scholar
  9. 9.
    Cação, I.: Constructive approximation by monogenic polynomials, Ph.D thesis, Univ. Aveiro (2004)Google Scholar
  10. 10.
    Cação I., Gürlebeck K., Bock S.: On derivatives of spherical monogenics. Complex Var. Elliptic Equ. 51(811), 847–869 (2006)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Cação I., Gürlebeck K., Bock S.: Complete orthonormal systems of spherical monogenics—a constructive approach. In: Son, L.H., Tutschke, W., Jain, S. (eds) Methods of Complex and Clifford Analysis, Proceedings of ICAM, Hanoi, SAS International Publications, Delhi (2004)Google Scholar
  12. 12.
    Cação I., Gürlebeck K., Malonek H.R.: Special monogenic polynomials and L 2-approximation. Adv. appl. Clifford Alg. 11(S2), 47–60 (2001)zbMATHCrossRefGoogle Scholar
  13. 13.
    Delanghe R., Lávička R., Souček V.: On polynomial solutions of generalized Moisil-Théodoresco systems and Hodge systems. Adv. appl. Clifford Alg. 21(3), 521–530 (2011)zbMATHCrossRefGoogle Scholar
  14. 14.
    Delanghe, R., Lávička, R., Souček, V.: The Fischer decomposition for Hodge-de Rham systems in Euclidean spaces (to appear in Math. Methods Appl. Sci., 2010). arXiv:1012.4994v1 [math.CV]Google Scholar
  15. 15.
    Delanghe, R., Lávička, R., Souček, V.: The Gelfand–Tsetlin bases for Hodge-de Rham systems in Euclidean spaces (to appear in Math. Methods Appl. Sci., 2010). arXiv:1012.4998v1 [math.CV]Google Scholar
  16. 16.
    Delanghe R., Sommen F., Souček V.: Clifford algebra and spinor-valued functions. Kluwer Academic Publishers, Dordrecht (1992)zbMATHCrossRefGoogle Scholar
  17. 17.
    Gelfand, I.M., Tsetlin, M.L.: Finite-dimensional representations of groups of orthogonal matrices (Russian), Dokl. Akad. Nauk SSSR 71 (1950), 1017–1020. English transl. in: I. M. Gelfand, Collected papers, Vol II. Springer, Berlin, pp. 657–661 (1988)Google Scholar
  18. 18.
    Gilbert J.E., Murray M.A.M.: Clifford Algebras and Dirac Operators in Harmonic Analysis. Cambridge University Press, Cambridge (1991)zbMATHCrossRefGoogle Scholar
  19. 19.
    Gürlebeck, K., Habetha, K., Sprößig, W.: Holomorphic functions in the plane and n-dimensional space. Translated from the 2006 German original, with cd-rom (Windows and UNIX). Birkhäuser, Basel (2008)Google Scholar
  20. 20.
    Gürlebeck K., Sprößig W.: Quaternionic and Clifford Calculus for Physicists and Engineers. Wiley, Chichester (1997)zbMATHGoogle Scholar
  21. 21.
    Lávička R.: Canonical bases for sl(2,C)-modules of spherical monogenics in dimension 3. Arch. Math.(Brno) 46(5), 339–349 (2010)MathSciNetGoogle Scholar
  22. 22.
    Lávička R.: The Fischer Decomposition for the H-action and Its Applications. In: Sabadini, I., Sommen, F. (eds) Hypercomplex analysis and applications. Trends in Mathematics, pp. 139–148. Springer, Basel (2011)CrossRefGoogle Scholar
  23. 23.
    Lávička, R., Souček, V., Van Lancker, P.: Orthogonal basis for spherical monogenics by step two branching. Ann. Global Anal. Geom. doi: 10.1007/s10455-011-9276-y (2011)
  24. 24.
    Molev A. I.: Gelfand-Tsetlin bases for classical Lie algebras. In: Hazewinkel, M. (ed) Handbook of Algebra, vol. 4, pp. 109–170. Elsevier, Amsterdam (2006)CrossRefGoogle Scholar
  25. 25.
    Sommen, F.: Spingroups and spherical means III, Rend. Circ. Mat. Palermo (2) Suppl. No 1 295–323 (1989)Google Scholar
  26. 26.
    Van Lancker P.: Spherical monogenics: an algebraic approach. Adv. Appl. Clifford Alg. 19, 467–496 (2009)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Faculty of Mathematics and PhysicsCharles UniversityPraha 8Czech Republic

Personalised recommendations