Advertisement

On Orthonormal Polynomial Solutions of the Riesz System in ℝ3

  • K. GürlebeckEmail author
  • J. Morais

Abstract

The main goal of the paper is to deal with a special orthogonal system of polynomial solutions of the Riesz system in ℝ3. The restriction to the sphere of this system is analogous to the complex case of the Fourier exponential functions {e in θ } n≥0 on the unit circle and has the additional property that also the scalar parts of the polynomials form an orthogonal system. The properties of the system will be applied to the explicit calculation of conjugate harmonic functions with a certain regularity.

Keywords

Quaternionic analysis Homogeneous monogenic polynomials Riesz system Harmonic conjugates 

Mathematics Subject Classification (2000)

30G35 31B05 31B35 

References

  1. 1.
    Abul-Ez, M.A., Constales, D.: Basic sets of polynomials in Clifford analysis. Complex Var. 14(1–4), 177–185 (1990) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Abul-Ez, M.A., Constales, D.: Linear substitution for basic sets of polynomials in Clifford analysis. Portugaliae Math. 48(2), 143–154 (1991) MathSciNetzbMATHGoogle Scholar
  3. 3.
    Abul-Ez, M.A., Constales, D.: The square root base of polynomials in Clifford analysis. Arch. Math. 80(5), 486–495 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Avetisyan, K., Gürlebeck, K., Sprößig, W.: Harmonic conjugates in weighted Bergman spaces of quaternion-valued functions. Comput. Methods Funct. Theory 9(2), 593–608 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Brackx, F., Delanghe, R.: On harmonic potential fields and the structure of monogenic functions. Z. Anal. Anwend. 22(2), 261–273 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Brackx, F., Delanghe, R., Sommen, F.: Clifford Analysis. Pitman, Boston/London/Melbourne (1982) zbMATHGoogle Scholar
  7. 7.
    Brackx, F., Delanghe, R., Sommen, F.: On conjugate harmonic functions in Euclidean space. Math. Methods Appl. Sci. 25(16–18), 1553–1562 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Brackx, F., De Schepper, N., Sommen, F.: Clifford algebra-valued orthogonal polynomials in Euclidean space. J. Approx. 137(1), 108–122 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Cação, I.: Constructive Approximation by Monogenic Polynomials. Ph.D. thesis, Universidade de Aveiro, Departamento de Matemática, Dissertation (2004) Google Scholar
  10. 10.
    Cação, I., Gürlebeck, K., Malonek, H.: Special monogenic polynomials and L 2-approximation. Adv. Appl. Clifford Algebras 11(S2), 47–60 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Cação, I., Gürlebeck, K., Bock, S.: On derivatives of spherical monogenics. Complex Var. Elliptic Equ. 51(8–11), 847–869 (2006) MathSciNetzbMATHGoogle Scholar
  12. 12.
    Cação, I., Gürlebeck, K., Bock, S.: Complete orthonormal systems of spherical monogenics—a constructive approach. In: Son, L.H., et al. (eds.) Methods of Complex and Clifford Analysis, pp. 241–260. SAS International, Delhi (2005) Google Scholar
  13. 13.
    Cnops, J.: Orthogonal functions associated with the Dirac operator. Ph.D. thesis, Ghent university (1989) Google Scholar
  14. 14.
    De Bie, H., Sommen, F.: Hermite and Gegenbauer polynomials in superspace using Clifford analysis. J. Phys. A, Math. Theor. 40(34), 10441–10456 (2007) zbMATHCrossRefGoogle Scholar
  15. 15.
    Delanghe, R.: Clifford analysis: history and perspective. Comput. Methods Funct. Theory 1(1), 107–153 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Delanghe, R.: On homogeneous polynomial solutions of the Riesz system and their harmonic potentials. Complex Var. Elliptic Equ. 52(10–11), 1047–1062 (2007) MathSciNetzbMATHGoogle Scholar
  17. 17.
    Delanghe, R.: On a class of inner spherical monogenics and their primitives. Adv. Appl. Clifford Algebras 18(3–4), 557–566 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Falcão, M.I., Malonek, H.: Generalized exponentials through Appell sets in ℝn+1 and Bessel functions. In: AIP-Proceedings, pp. 738–741 (2007) Google Scholar
  19. 19.
    Falcão, M.I., Malonek, H.: Special monogenic polynomials—properties and applications. In: AlP-Proceedings, pp. 764–767 (2007) Google Scholar
  20. 20.
    Falcão, M., Cruz, J., Malonek, H.: Remarks on the generation of monogenic functions. In: Gürlebeck, K., Könke, C. (eds.) Proceedings 17th International Conference on the Applications of Computer Science and Mathematics in Architecture and Civil Engineering. Weimar (2006) Google Scholar
  21. 21.
    Fueter, R.: Analytische Funktionen einer Quaternionenvariablen. Comment. Math. Helv. 4, 9–20 (1932) MathSciNetCrossRefGoogle Scholar
  22. 22.
    Fueter, R.: Functions of a Hyper Complex Variable. Lecture notes written and supplemented by E. Bareiss, Math. Inst. Univ. Zürich, Fall Semester (1949) Google Scholar
  23. 23.
    Gürlebeck, K.: Interpolation and best approximation in spaces of monogenic functions. Wiss. Z. TU Karl-Marx-Stadt 30, 38–40 (1988) Google Scholar
  24. 24.
    Gürlebeck, K., Malonek, H.: A hypercomplex derivative of monogenic functions in ℝn+1 and its applications. Complex Var. Elliptic Equ. 39(3), 199–228 (1999) zbMATHCrossRefGoogle Scholar
  25. 25.
    Gürlebeck, K., Morais, J.: On mapping properties of monogenic functions. CUBO A Math. J. 11(1), 73–100 (2009) zbMATHGoogle Scholar
  26. 26.
    Gürlebeck, K., Morais, J.: Bohr type theorems for monogenic power series. Comput. Methods Funct. Theory 9(2), 633–651 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Gürlebeck, K., Morais, J.: On local mapping properties of monogenic functions. In: Gürlebeck, K., Könke, C. (eds.) Proceedings 18th International Conference on the Applications of Computer Science and Mathematics in Architecture and Civil Engineering. Weimar (2009) Google Scholar
  28. 28.
    Gürlebeck, K., Sprössig, W.: Quaternionic Analysis and Elliptic Boundary Value Problems. Akademie Verlag, Berlin (1989) zbMATHGoogle Scholar
  29. 29.
    Gürlebeck, K., Sprössig, W.: On the treatment of fluid problems by methods of Clifford analysis. Math. Comput. Simul. 44(4), 401–413 (1997) zbMATHCrossRefGoogle Scholar
  30. 30.
    Gürlebeck, K., Sprössig, W.: Quaternionic Calculus for Engineers and Physicists. Wiley, Chichester (1997) zbMATHGoogle Scholar
  31. 31.
    Gürlebeck, K., Sprössig, W.: On eigenvalue estimates of nonlinear Stokes eigenvalue problems. In: Micali, A., et al. (eds.) Clifford Algebras and Their Applications in Mathematical Physics, pp. 327–333. Kluwer Academic, Amsterdam (1992) Google Scholar
  32. 32.
    Kamzolov, A.: The best approximation of the classes of functions \(W^{\alpha }_{p}(S^{n})\) by polynomials in spherical harmonics. Math. Not. 32(3), 622–626 (1982) MathSciNetCrossRefGoogle Scholar
  33. 33.
    Kravchenko, V.: Applied Quaternionic Analysis. Research and Exposition in Mathematics, vol. 28. Heldermann, Lemgo (2003) zbMATHGoogle Scholar
  34. 34.
    Kravchenko, V., Shapiro, M.: Integral Representations for Spatial Models of Mathematical Physics. Research Notes in Mathematics. Pitman Advanced Publishing Program, London (1996) zbMATHGoogle Scholar
  35. 35.
    Leutwiler, H.: Quaternionic analysis in ℝ3 versus its hyperbolic modification. In: Brackx, F., Chisholm, J.S.R., Soucek, V. (eds.) NATO Science Series II. Mathematics, Physics and Chemistry, vol. 25. Kluwer Academic, Dordrecht/Boston/London (2001) Google Scholar
  36. 36.
    Malonek, H.: Power series representation for monogenic functions in ℝm+1 based on a permutational product. Complex Var. Elliptic Equ. 15(3), 181–191 (1990) MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Mitelman, I., Shapiro, M.: Differentiation of the Martinelli-Bochner integrals and the notion of hyperderivability. Math. Nachr. 172(1), 211–238 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Moisil, G.: Sur la généralisation des fonctions conjuguées. Rend. Acad. Naz. Lincei 14, 401–408 (1931) Google Scholar
  39. 39.
    Morais, J.: Approximation by homogeneous polynomial solutions of the Riesz system in ℝ3. Ph.D. thesis, Bauhaus-Universität Weimar (2009) Google Scholar
  40. 40.
    Riesz, M.: Clifford numbers and spinors. Inst. Phys. Sci. Techn. Lect. Ser., vol. 38. Maryland (1958) Google Scholar
  41. 41.
    Ryan, J.: Left regular polynomials in even dimensions, and tensor products of Clifford algebras. In: Chisholm, J.S.R., Common, A.K. (eds.) Clifford Algebras and Their Applications in Mathematical Physics, pp. 133–147. Reidel, Dordrecht (1986) CrossRefGoogle Scholar
  42. 42.
    Sansone, G.: Orthogonal Functions. Pure and Applied Mathematics, vol. IX. Interscience Publishers, New York (1959) zbMATHGoogle Scholar
  43. 43.
    Shapiro, M., Vasilevski, N.L.: Quaternionic ψ-hyperholomorphic functions, singular operators and boundary value problems I. Complex Var. Theory Appl. (1995) Google Scholar
  44. 44.
    Shapiro, M., Vasilevski, N.L.: Quaternionic ψ-hyperholomorphic functions, singular operators and boundary value problems II. Complex Var. Theory Appl. (1995) Google Scholar
  45. 45.
    Sprössig, W.: Boundary value problems treated with methods of Clifford analysis. Contemp. Math. 212, 255–268 (1998) CrossRefGoogle Scholar
  46. 46.
    Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970) zbMATHGoogle Scholar
  47. 47.
    Stein, E.M., Weiß, G.: On the theory of harmonic functions of several variables. Part I: the theory of H p spaces. Acta Math. 103, 25–62 (1960) MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Sudbery, A.: Quaternionic analysis. Math. Proc. Camb. Philos. Soc. 85, 199–225 (1979) MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Whittaker, J.M.: Sur les séries de base de polynômes quelconques. Avec la col laboration de C. Gattegno (Collection de monographies sur la theorie des fonctions). Paris: Gauthier-Villars (1949) Google Scholar
  50. 50.
    Xu, Z.: Boundary value problems and function theory for spin-invariant differential operators. Ph.D thesis. Gent (1989) Google Scholar
  51. 51.
    Xu, Z., Zhou, C.: On boundary value problems of Riemann-Hilbert type for monogenic functions in a half space of ℝm (m≥2). Complex Var. Theory Appl. 22, 181–194 (1993) MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Xu, Z., Chen, J., Zhang, W.: A harmonic conjugate of the Poisson kernel and a boundary value problem for monogenic functions in the unit ball of ℝn (n≥2). Simon Stevin 64, 187–201 (1990) MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Institut für Mathematik/PhysikBauhaus-Universität WeimarWeimarGermany

Personalised recommendations