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.
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Andrews G.E., Askey R., Roy R.: Special Functions, Encyclopedia of Mathematics and its Applications, vol. 71. Cambridge University Press, Cambridge (1999)
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)
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)
Bock S., Gürlebeck K.: On a generalized Appell system and monogenic power series. Math. Methods Appl. Sci. 33, 394–411 (2010)
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]
Brackx F., Delanghe R., Sommen F.: Clifford analysis. Pitman, London (1982)
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)
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]
Cação, I.: Constructive approximation by monogenic polynomials, Ph.D thesis, Univ. Aveiro (2004)
Cação I., Gürlebeck K., Bock S.: On derivatives of spherical monogenics. Complex Var. Elliptic Equ. 51(811), 847–869 (2006)
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)
Cação I., Gürlebeck K., Malonek H.R.: Special monogenic polynomials and L 2-approximation. Adv. appl. Clifford Alg. 11(S2), 47–60 (2001)
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)
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]
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]
Delanghe R., Sommen F., Souček V.: Clifford algebra and spinor-valued functions. Kluwer Academic Publishers, Dordrecht (1992)
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)
Gilbert J.E., Murray M.A.M.: Clifford Algebras and Dirac Operators in Harmonic Analysis. Cambridge University Press, Cambridge (1991)
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)
Gürlebeck K., Sprößig W.: Quaternionic and Clifford Calculus for Physicists and Engineers. Wiley, Chichester (1997)
Lávička R.: Canonical bases for sl(2,C)-modules of spherical monogenics in dimension 3. Arch. Math.(Brno) 46(5), 339–349 (2010)
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)
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)
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)
Sommen, F.: Spingroups and spherical means III, Rend. Circ. Mat. Palermo (2) Suppl. No 1 295–323 (1989)
Van Lancker P.: Spherical monogenics: an algebraic approach. Adv. Appl. Clifford Alg. 19, 467–496 (2009)
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Communicated by Fred Brackx.
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Lávička, R. Complete Orthogonal Appell Systems for Spherical Monogenics. Complex Anal. Oper. Theory 6, 477–489 (2012). https://doi.org/10.1007/s11785-011-0200-z
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DOI: https://doi.org/10.1007/s11785-011-0200-z
Keywords
- Spherical harmonics
- Spherical monogenics
- Gelfand–Tsetlin basis
- Appell system
- Orthogonal basis
- Taylor series