Clifford Analysis and Harmonic Polynomials

  • Klaus Gürlebeck
  • Wolfgang Sprößig
Reference work entry


This overview gives an insight in the new field of hypercomplex analysis in relation to harmonic analysis. The algebra of complex numbers is replaced by the non-commutative algebra of real quaternions or by Clifford algebras. This contribution is focused on the presentation of an operator calculus on the sphere as well as the discussion of monogenic and holomorphic orthonormal systems of polynomials and special functions on the unit ball and the sphere. Function theoretic elements are lined out. Relations to the Lie algebra SO(3) are discussed and a corresponding Radon transform is introduced.


  1. Appell P (1880) Sur une class de polynomes. Annales Scientifiques de l‘Ecole Normale Suprieure 9:119–144Google Scholar
  2. Avetisyan K, Guerlebeck K, Sprößig W (2009) Harmonic conjugates in weighted Bergman spaces of quaternion-valued functions. Comput Methods Funct Theory 9(2):593–608MathSciNetCrossRefMATHGoogle Scholar
  3. Axler S, Bourdon P, Ramey W (2001) Harmonic function theory, 2nd edn. Springer, New YorkCrossRefMATHGoogle Scholar
  4. Bock S (2012) On a three dimensional analogue to the holomorphic z-powers: power series and recurrence formulae. Complex Var Elliptic Equ Int J 57(12):1349–1370MathSciNetCrossRefMATHGoogle Scholar
  5. Brackx F, Delanghe R, Sommen F (1982) Clifford analysis. Pitman research notes in mathematics, vol 76. Pitman, LondonMATHGoogle Scholar
  6. Cacao I (2004) Constructive approximation by monogenic polynomials. Ph.D. University of AveiroMATHGoogle Scholar
  7. Cerejeiras P, Schaeben H, Sommen F (2002) The spherical X-ray transform. In: Sommen F, Sprössig W (eds) Clifford analysis in application. Mathematical methods in the applied sciences, vol 25, pp 1493–1507. John Wiley & Sons, ChichesterGoogle Scholar
  8. Chisholm M (2002) Such silver currents: the story of William and Lucy Clifford, 1845–1929. Lutterworth Press, CambridgeGoogle Scholar
  9. Clifford WK (1878) Applications of Grassmann’s extensive algebra. Am J Math Pure Appl 1: 350–358MathSciNetMATHGoogle Scholar
  10. Cnops J (2002) An introduction of Dirac operators on manifolds. Birkhäuser, BaselCrossRefMATHGoogle Scholar
  11. Cockle J (1848) On certain functions resembling quaternions and on a new imaginary in algebra. Lond-Dublin-Edinb Philos Mag Ser 3, 33:435–439Google Scholar
  12. Crowe MC (1967) A history of vector analysis. Dover, New YorkMATHGoogle Scholar
  13. Delanghe R (2007) On homogeneous polynomial solutions of the Riesz system and their harmonic potentials. Complex Var Elliptic Equ 52(10–11):1047–1062MathSciNetCrossRefMATHGoogle Scholar
  14. Delanghe R, Sommen F, Soucek V (1992) Clifford algebra and spinor valued functions. Kluwer, DordrechtCrossRefMATHGoogle Scholar
  15. Duduchava LR, Mitrea D, Mitrea M (2006) Differential operators and boundary value problems on hypersurfaces. Math Nachr 279(9–10):996–1023MathSciNetCrossRefMATHGoogle Scholar
  16. Euler L (1775) Nova methodus motum corporum rigidorum determinandi. Novi Comment Acad Sci Imperalis Petropolitanae 20:208–238. Reprinted in pp. 99–125 of Euler L (1968) Leonhardi Euleri Opera Omnia, II, vol 9. Orell Füssli, Zürich (1968). Edited by C BlancGoogle Scholar
  17. Fengler MJ, Freeden W (2005) A non-linear Galerkin scheme involving vector and tensor spherical harmonics for solving the incompressible Navier-Stokes equations on the sphere. SIAM J Sci Comput 27:967–994MathSciNetCrossRefMATHGoogle Scholar
  18. Freeden W, Gervens T, Schreiner M (1998) Constructive approximation on the sphere: with application to geomathematics. Numerical mathematics and scientific computation. Clarendon, OxfordMATHGoogle Scholar
  19. Fueter R (1935/1936) Über die analytische Darstellung der regulären Funktionen einer Quaternionenvariablen. Comment Math Helv 8:371–378MathSciNetCrossRefMATHGoogle Scholar
  20. Gauss CF (1900) Mutation des Raumes. In: Stöckl (ed) Carl Friedrich Gauss Werke, Achter Band. Koniglichen Gesellschaft der Wissenschaften, Göttingen, pp 357–361Google Scholar
  21. Gibbs JW (1881) Elements of vector analysis. Tuttle, Morehouse Taylor, New-HavenMATHGoogle Scholar
  22. Grassmann HG (1844) Die Wissenschaft der extensiven Größe oder die Ausdehnungslehre, eine neue mathematische Disziplin, 1. Teil: Die lineale Ausdehnungslehre, Leipzig. (Reprint 1878) (GW 1,1, S. 4–312)Google Scholar
  23. Gürlebeck K, Morais J (2009a) On mapping properties of monogenic functions. CUBO A Math J 11(1):73–100MathSciNetMATHGoogle Scholar
  24. Gürlebeck K, Morais J (2009b) Bohr type theorem for monogenic power series. Comput Methods Funct Theory 9(2):633–651MathSciNetCrossRefMATHGoogle Scholar
  25. Gürlebeck K, Sprößig W (1997a) Quaternionic and Clifford calculus for physicists and engineers. Wiley, ChichesterMATHGoogle Scholar
  26. Gürlebeck K, Sprößig W (1997b) On the treatment of fluid flow problems by methods of Clifford analysis. Math Comput Simul 44(4):401–413CrossRefMATHGoogle Scholar
  27. Gürlebeck K, Sprößig W (2010) Fluid flow problems with quaternion analysis – an alternative conception. In: Bayro-Corrochano E, Scheuermann G (eds) Geometric algebraic computing in engineering and computer science. Springer, London, pp 345–380CrossRefGoogle Scholar
  28. Gürlebeck K, Habetha K, Sprößig W (2008) Holomorphic functions in the plane and n-dimensional space. Birkhäuser, BaselMATHGoogle Scholar
  29. Hamilton WR (1866) Elements of quaternions. Longmans Green, London. Reprinted by Chelsea, New York (1969)Google Scholar
  30. Hardy GH, Littlewood JE (1931) Some properties of conjugate functions. J Reine Angew Math 167:405–423MathSciNetMATHGoogle Scholar
  31. Hielscher R (2007) The radon transform on the rotation group – inversion and application to texture analysis. Thesis, FreibergGoogle Scholar
  32. Hielscher R, Mainprice D, Schaeben H (2010) Material behavior: texture and anisotropy. In: Freeden W, Nashed MZ (eds) Handbook of geomathematics. Springer, Berlin/Heidelberg, pp 974–1000Google Scholar
  33. Hochstadt H (1971) The functions of mathematical physics. Wiley, New YorkMATHGoogle Scholar
  34. Krylov NM (1947) Sur les quaternions de W.R.Hamilton et la notion de la monogenique. Dokl Acad Nauk SSSR 55:787–788Google Scholar
  35. Lam TY (2003) Hamilton’s quaternions. Handbook of algebra, vol 3. North-Holland, Amsterdam, pp 429–454Google Scholar
  36. Leutwiler L (2001) Quaternionic analysis in \(\mathbb{R}^{3}\) versus its hyperbolic modification. In: Brackx F et al (eds) Clifford analysis and its applications. Kluwer, Dordrecht, pp 193–211CrossRefGoogle Scholar
  37. Majda A (2003) Introduction to PDEs and waves for the atmosphere and ocean. Courant lecture notes in mathematics, vol 9. American Mathematical Society (AMS)/Courant Institute of Mathematical Sciences, Providence/New YorkGoogle Scholar
  38. Malonek HR (1987) Zum Holomorphiebegriff in höheren Dimensionen. Habilitationsschrift, PH HalleGoogle Scholar
  39. Malonek HR (1990) A new hypercomplex structure of the Euclidean space \(\mathbb{R}^{n+1}\) and a concept of hypercomplex differentiability. Complex Var 14:25–33MathSciNetCrossRefMATHGoogle Scholar
  40. Malonek HR (1993) Hypercomplex differentiability and its applications. In: Brackx F et al (eds) Clifford algebras and applications in mathematical physics. Kluwer, Dordrecht, pp 141–150CrossRefGoogle Scholar
  41. Mejlikhzhon AS (1948) On the notion of monogeneous quaternions (Russian). Dokl Akad Nauk SSSR 59:431–434Google Scholar
  42. Mitrea M (2002) Boundary value problems for Dirac operators and Maxwell’s equations in non-smooth domains. MMAS 25:1355–1369MathSciNetMATHGoogle Scholar
  43. Mitelman IM, Shapiro MV (1995) Differentiation of the Martinelli-Bochner integrals and the notion of hyperderivability. Math Nachr 172:211–238MathSciNetCrossRefMATHGoogle Scholar
  44. Morais J (2009) Approximation by homogeneous polynomial solutions of the Riesz system in \(\mathbb{R}^{3}\). Ph.D. diss., Bauhaus-University WeimarGoogle Scholar
  45. Qian T, Yang Y (2009) Hilbert transforms on the sphere with the Clifford algebra setting. J Fourier Anal Appl 15:753–774MathSciNetCrossRefMATHGoogle Scholar
  46. Qian T, Sprößig W, Wang J (2012) Adaptive Fourier decomposition of functions in the orthogonal rational system of quaternionic values. Math Methods Appl Sci 35(1):43–64MathSciNetCrossRefMATHGoogle Scholar
  47. Rodrigues O (1840) Des lois geométriques qui régissent les déplacements d’un systéme solide dans l’espace, et de la variation des coordonnées provenant de produire. J Math Pures Appl 5:380–440Google Scholar
  48. Porteous I (1969) Topological geometry. Van Nostrand-Reinhold, LondonMATHGoogle Scholar
  49. Sansone G (1959) Orthogonal functions. Pure and applied mathematics, vol 9. Interscience, New YorkGoogle Scholar
  50. Schaeben H, Sprößig W, Van den Boogaart B (2000) The spherical X-Ray transform of texture goniometry. In: Brackx F, Chisholm JSR, Soucek V (eds) Clifford analysis and its applications. Proceedings of NATO advanced research workshop on Clifford analysis and applications, Prague, 30 Oct–3 Nov 2000, pp 283–291Google Scholar
  51. Sprößig W (2011) Forecasting equations in complex quaternionic setting. In: Simos, Theodore E. (ed), Recent advances in computational and applied mathematics. Dordrecht: SpringerGoogle Scholar
  52. Sprößig W, Fichtner A (2004) Eagle-Guide: Vektoranalysis. Edition am Gutenbergplatz, LeipzigGoogle Scholar
  53. Sudbery A (1979) Quaternionic analysis. Math Proc Camb Philos Soc 85:199–225MathSciNetCrossRefMATHGoogle Scholar
  54. Vilenkin NJ, Klimyk AU (1991) Representation of lie groups and special functions, vol 1. Kluwer Academic, DordrechtCrossRefMATHGoogle Scholar
  55. Van Lancker P (1997) Clifford analysis on the unit sphere. Thesis, University of GhentGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Bauhaus-Universität WeimarWeimarGermany
  2. 2.TU Bergakademie FreibergFreibergGermany

Personalised recommendations