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Computational Methods and Function Theory

, Volume 12, Issue 2, pp 501–515 | Cite as

Generalized Derivative and Primitive of Cliffordian Bases of Polynomials Constructed Through Appell Monomials

  • Mohra ZayedEmail author
  • Mahmoud Abul-Ez
  • João Pedro Morais
Article

Abstract

In the present paper the authors treat two different problems. They start by answering the question posed in [5] concerning the structure of derivative bases, then investigate the convergence properties (the effectiveness) of the generalized derivative and primitive of a given base of polynomials with values in a Clifford algebra. Finally, they study the mode of increase of such derivatives and primitive bases.

En]Keywords

special monogenic polynomials effectiveness mode of increase 

2000 MSC

30G35 41A10 

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References

  1. 1.
    M. Abul-Ez and D. Constales, Basic sets of polynomials in Clifford analysis, Complex Variables 14 no.1-4(1990), 177–185.MathSciNetzbMATHGoogle Scholar
  2. 2.
    M. Abul-Ez and D. Constales, Linear substitution for basic sets of polynomials in Clifford analysis, Port. Math. 48 no.2 (1991), 143–154.MathSciNetzbMATHGoogle Scholar
  3. 3.
    M. Abul-Ez and D. Constales, The square root base of polynomials in Clifford analysis, Arch. Math 80 no.5 (2003), 486–495.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    M. Abul-Ez and D. Constales, On convergence properties of basic series representing special monogenic functions, Arch. Math. 81 no.1 (2002), 62–71.MathSciNetCrossRefGoogle Scholar
  5. 5.
    L. Aloui and G. Hassan, Hypercomplex derivative bases of polynomials in Clifford analysis, Math. Methods Appl. Sci. 33 no.3 (2009), 350–357.MathSciNetGoogle Scholar
  6. 6.
    P. Appell, Sur une classe de pôlynomes, Ann. Sci. Ecole Norm. Sup. (9) 21 (1880).Google Scholar
  7. 7.
    S. Bock and K. Gürlebeck, On a generalized Appell system and monogenic power series, Math. Methods Appl. Sci. 33 no.4 (2009), 394–411.Google Scholar
  8. 8.
    F. Brackx, R. Delanghe and F. Sommen, Clifford Analysis, Pitman, Boston, London, Melbourne, 1982.zbMATHGoogle Scholar
  9. 9.
    I. Cação, Constructive approximation by monogenic polynomials, Ph.D. thesis, Universidade de Aveiro, 2004.Google Scholar
  10. 10.
    I. Cação, Complete orthonormal sets of polynomial solutions of the Riesz and Moisil-Teodorescu systems in ℝ3, Numer. Algor. 55 no.2-3 (2010), 191–203.zbMATHCrossRefGoogle Scholar
  11. 11.
    I. Cação and H. Malonek, On complete sets of hypercomplex Appell polynomials, in: T.E. Simos, G. Psihoyios, Ch. Tsitouras (eds.), Numerical Analysis and Applied Mathematics, AIP Conference Proceedings 1048, American Institute of Physics: Melville, NY, (2008), 647–650.Google Scholar
  12. 12.
    D. Constales, R. De Almeida and R. S. Kraußhar, On the growth type of entire monogenic functions, Arch. Math. 88 no.2 (2007), 153–163.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    R. Delanghe, R. Lávička and V. Soucek, On polynomial solutions of generalized Moisil-Théodoresco systems and Hodge systems, Adv. Appl. Clifford Alg. 21 no.3 (2011), 521–530.zbMATHCrossRefGoogle Scholar
  14. 14.
    D. Peña Peña, On a sequence of monogenic polynomials satisfying the Appell condition whose first term is a non-constant function, arXiv:1102.1833 [math.CV] (2011), submitted.Google Scholar
  15. 15.
    —, Shifted Appell sequences in Clifford analysis, arXiv:1102.4373 [math.CV] (2011), submitted.Google Scholar
  16. 16.
    M. Falcão, J. Cruz and H. Malonek, Remarks on the generation of monogenic functions, in: K. Gürlebeck and C. Könke (eds.) Proceedings 17th International Conference on the Applications of Computer Science and Mathematics in Architecture and Civil Engineering, Weimar, 2006.Google Scholar
  17. 17.
    M. Falcão and H. Malonek, Generalized exponentials through Appell sets in ℝn+1 and Bessel functions, AIP-Proceedings (2007), 738–741.Google Scholar
  18. 18.
    —, Special monogenic polynomials — properties and applications, AlP-Proceedings (2007), 764–767.Google Scholar
  19. 19.
    R. Fueter, Analytische Funktionen einer Quaternionenvariablen, Comment. Math. Helv. 4 (1932), 9–20.MathSciNetCrossRefGoogle Scholar
  20. 20.
    R. Fueter, Functions of a Hypercomplex Variable, lecture notes written and supplemented by E. Bareiss, Math. Inst. Univ. Zürich, Fall Semester, 1949.Google Scholar
  21. 21.
    K. Gürlebeck and W. Sprössig, Quaternionic and Clifford Calculus for Engineers and Physicists, John Wiley and Sons, Chichester, 1997.zbMATHGoogle Scholar
  22. 22.
    K. Gürlebeck and W. Sprössig, On the treatment of fluid problems by methods of Clifford analysis, Math. Comput. Simulation 44 no.4 (1997), 401–413.MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    K. Gürlebeck and H. Malonek, A hypercomplex derivative of monogenic functions in ℝn+1 and its applications, Complex Variables 39 no.3 (1999), 199–228.MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    K. Gürlebeck, K. Habetha and W. Sprößig, Holomorphic Functions in the Plane and n-Dimensional Space, Birkhäuser Verlag, Basel/Boston/Berlin, 2008.zbMATHGoogle Scholar
  25. 25.
    N. Gürlebeck, On Appell Sets and the Fueter-Sce Mapping, Adv. Appl. Clifford Algebras 19 no.1 (2009), 51–61.zbMATHCrossRefGoogle Scholar
  26. 26.
    R. Lávicka, Generalized Appell property for the Riesz system in dimension 3, AIP Conference Proceedings 1389 (2011), 291–294.CrossRefGoogle Scholar
  27. 27.
    —, Complete orthogonal Appell systems for spherical monogenics, arXiv:1106.2970v1 [math.CV] (2011), submitted.Google Scholar
  28. 28.
    E. Lindelöf, Sur la détermination de la croissance des fonctions enti`eres deéfinies par un développement de Taylor, Darb. Bull. 27 no.2 (1903), 213–226.Google Scholar
  29. 29.
    H. Malonek, Power series representation for monogenic functions in ℝm+1 based on a permutational product, Complex Var. 15 no.3 (1990), 181–191.MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    J. Morais and H.T. Le, Orthogonal Appell Systems of Monogenic Functions in the Cylinder, Math. Methods Appl. Sci. 34 no.12 (2011), 1472–1486.MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    V. Ozegov, Some extremal properties of generalized Appell polynomials, Soviet Math. 5 (1964), 1651–1653.Google Scholar
  32. 32.
    A. Pringsheim, Elementare Theorie der ganzen transzendenten Funktionen von endlicher Ordnung, Math. Ann. 58 (1904), 257–342.MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    I. Sheffer, Some properties of polynomial sets of type zero, Duke Math. J. 5 (1939), 590–622.MathSciNetCrossRefGoogle Scholar
  34. 34.
    I. Sheffer, Some applications of certain polynomial classes, Bull. Amer. Math. Soc. 47 no.12 (1941), 885–898.MathSciNetCrossRefGoogle Scholar
  35. 35.
    I. Sheffer, Note on Appell polynomials, Bull. Amer. Math. Soc. 51 no.10 (1945), 739–744.MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    C. Thorne, A property of Appell sets, Amer. Math. Monthly 52 (1945), 191–193.MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    G. Valiron, Lectures on the General Theory of Integral Functions, Chelsea, New York, 1949.Google Scholar
  38. 38.
    A. Wiman, Über den Zusammenhang zwischen dem Maximalbetrage einer analytischen Funktion und dem größten Gliede der zugehörigen Taylorschen Reihe, Acta Math. 37 (1914), 305–326.MathSciNetCrossRefGoogle Scholar
  39. 39.
    M. Webster, Orthogonal polynomials with orthogonal derivatives, Bull. Amer. Math. Soc. 44 (1938), 880–888.MathSciNetCrossRefGoogle Scholar
  40. 40.
    J. Whittaker, The uniqueness of expansions in polynomials, J. London Math. Soc. 10 (1935), 108–111.MathSciNetCrossRefGoogle Scholar
  41. 41.
    J. Whittaker, 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. VI., 1949.Google Scholar

Copyright information

© Heldermann  Verlag 2012

Authors and Affiliations

  • Mohra Zayed
    • 1
    Email author
  • Mahmoud Abul-Ez
    • 2
  • João Pedro Morais
    • 3
  1. 1.Department of MathematicsKing Khalid UniversityAbhaSaudi Arabia
  2. 2.Department of Mathematics, Faculty of ScienceSohag UniversitySohagEgypt
  3. 3.Institute of Applied AnalysisFreiberg University of Mining and TechnologyFreibergGermany

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