Advertisement

Results in Mathematics

, Volume 43, Issue 1–2, pp 1–12 | Cite as

Asymptotic approximation of functions and their derivatives by Müller’s Gamma operators

  • Ulrich Abel
  • Mircea Ivan
Article

Abstract

We obtain the complete asymptotic expansion of the image functions of Müller’s Gamma operators and of their derivatives. All expansion coefficients are explicitly calculated. Moreover, we study linear combinations of Gamma operators having a better degree of approximation than the operators themselves. Using divided differences we define general classes of linear combinations of which special cases were recently introduced and investigated by other authors.

Keywords

divided difference positive operator Stirling numbers 

MSC2000

41A36 11B73 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    U. Abel, The moments for the Meyer-König and Zeller operators, J. Approx. Theory 82 (1995), 352–361.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    U. Abel, On the asymptotic approximation with operators of Bleimann, Butzer and Hahn, Indag. Math., New Ser., 7 (1996), 1–9.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    U. Abel, The complete asymptotic expansion for the Meyer-König and Zeller operators, J. Math. Anal. Appl. 208 (1997), 109–119.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    U. Abel, Asymptotic approximation with Stancu Beta operators, Rev. Anal. Numér. Theor. Approx. 27: 1 (1998), 5–13.MathSciNetMATHGoogle Scholar
  5. [5]
    U. Abel, Asymptotic approximation with Kantorovich polynomials, Approx. Theory Appl. (N.S.) 14:3 (1998), 106–116.MathSciNetMATHGoogle Scholar
  6. [6]
    U. Abel, On the asymptotic approximation with bivariate operators of Bleimann, Butzer and Hahn, J. Approx. Theory 97 (1999), 181–198.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    U. Abel, On the asymptotic approximation with bivariate Meyer-König and Zeller operators, submitted. [8] U. Abel, Asymptotic approximation by Bernstein-Durrmeyer operators and their derivatives, Approx. Theory Appl. (N.S.) 16:2 (2000), 1–12.MathSciNetMATHGoogle Scholar
  8. [9]
    U. Abel and B. Delia Vecchia, Asymptotic approximation by the operators of K. Balázs and Szabados, Acta Sci. Math. (Szeged) 66 (2000), 137–145.MathSciNetMATHGoogle Scholar
  9. [10]
    U. Abel and M. Ivan, Asymptotic expansion of the multivariate Bernstein polynomials on a simplex, Approx. Theory Appl. (N.S.) 16:3 (2000), 85–93.MathSciNetMATHGoogle Scholar
  10. [11]
    U. Abel and M. Ivan, Asymptotic approximation with a sequence of positive linear operators, J. Comput. Anal. Appl., 3:4 (2001), 331–341.MathSciNetGoogle Scholar
  11. [12]
    L. Comtet, “Advanced Combinatorics”, Reidel Publishing Comp., Dordrecht, 1974.MATHCrossRefGoogle Scholar
  12. [13]
    C. Jordan, “Calculus of finite differences”, Chelsea, New York, 1965.MATHGoogle Scholar
  13. [14]
    A. Lupaş and M. W. Müller, Approximationseigenschaften der Gammaoperatoren, Math. Zeitschr. 98 (1967), 208–226.MATHCrossRefGoogle Scholar
  14. [15]
    A. Lupaş, D. H. Mache and M. W. Müller, Weighted L p-approximation of derivatives by the method of Gammaoperators, Results in Mathematics 28 (1995), 277–286.MATHCrossRefGoogle Scholar
  15. [16]
    A. Lupaş, D. H. Mache, V. Maier and M. W. Müller, Linear combinations of Gammaoperators in L p-spaces, Results in Mathematics 34 (1998), 156–168.MATHCrossRefGoogle Scholar
  16. [17]
    A. Lupaş, D. H. Mache, V. Maier and M. W. Müller, Certain results involving Gammaoperators, in: ”New Developments in Approximation Theory” (International Series of Numerical Mathematics, Vol. 132), (M. W. Müller, M. Buhmann, D. H. Mache, and M. Feiten, eds.) Birkhäuser-Verlag, Basel 1998, pp. 199–214.Google Scholar
  17. [18]
    M. W. Müller, “Die Folge der Gammaoperatoren”, Dissertation, Stuttgart, 1967.Google Scholar
  18. [19]
    M. W. Müller, Punktweise und gleichmäßige Approximation durch Gammaoperatoren, Math. Zeitschr. 103 (1968), 227–238.MATHCrossRefGoogle Scholar
  19. [20]
    M. W. Müller, The central approximation theorems for the method of left gamma quasi-interpolants in L p spaces, J. Comput. Anal Appl. 3 (2001), 207–222.MathSciNetMATHGoogle Scholar
  20. [21]
    P. Sablonniére, Representation of quasi-interpolants as differential operators and applications, in: “New Developments in Approximation Theory” (International Series of Numerical Mathematics, Vol. 132), (M. W. Müller, M. Buhmann, D. H. Mache, and M. Feiten, eds.) Birkhäuser-Verlag, Basel 1998, pp. 233–253.Google Scholar
  21. [22]
    P. C. Sikkema, On some linear positive operators, Indag. Math. 32 (1970), 327–337.MathSciNetGoogle Scholar
  22. [23]
    P. C. Sikkema, On the asymptotic approximation with operators of Meyer-König and Zeller, Indag. Math. 32 (1970), 428–440.MathSciNetGoogle Scholar
  23. [24]
    V. Totik, The gammaoperators in L p spaces, Publ. Math. Debrecen 32 (1985), 43–55.MathSciNetMATHGoogle Scholar

Copyright information

© Birkhäuser Verlag, Basel 2003

Authors and Affiliations

  1. 1.Fachbereich MND, Fachhochschule Giessen-FriedbergUniversity of Applied SciencesFriedbergGermany
  2. 2.Department of MathematicsTechnical University of Cluj-NapocaCluj-NapocaRomania

Personalised recommendations