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Numerical Algorithms

, Volume 44, Issue 1, pp 69–82 | Cite as

The approximation of logarithmic function by q-Bernstein polynomials in the case q > 1

  • Sofiya Ostrovska
Original Paper

Abstract

Since in the case q > 1, q-Bernstein polynomials are not positive linear operators on C[0,1], the study of their approximation properties is essentially more difficult than that for 0<q<1. Despite the intensive research conducted in the area lately, the problem of describing the class of functions in C[0,1] uniformly approximated by their q-Bernstein polynomials (q > 1) remains open. It is known that the approximation occurs for functions admit ting an analytic continuation into a disc {z:|z| < R}, R > 1. For functions without such an assumption, no general results on approximation are available. In this paper, it is shown that the function f(x) = ln (x + a), a > 0, is uniformly approximated by its q-Bernstein polynomials (q > 1) on the interval [0,1] if and only if a ≥ 1.

Keywords

q-integers q-binomial coefficients q-Bernstein polynomials Uniform convergence 

Mathematics Subject Classifications (2000)

41A10 30E10 

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References

  1. 1.
    Bernstein, S.N.: Démonstration du théorème de Weierstrass fondée sur la calcul des probabilités. Comm. Soc. Math. Charkow t.13(série 2), 1–2 (1912)Google Scholar
  2. 2.
    Cooper, S., Waldron, S.: The eigenstructure of the Bernstein operator. J. Approx. Theory 105, 133–165 (2000)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Derriennic, M.-M.: Modified Bernstein polynomials and Jacobi polynomials in q-calculus. Rend. Circ. Mat. Palermo Suppl. 76(Serie II), 269–290 (2005)MathSciNetGoogle Scholar
  4. 4.
    DeVore, R., Lorentz, G.G.: Constructive Approximation. Springer, Berlin Heidelberg New York (1993)MATHGoogle Scholar
  5. 5.
    Goodman, T.N.T., Oruç, H., Phillips, G.M.: Convexity and generalized Bernstein polynomials. Proc. Edinb. Math. Soc. 42(1), 179–190 (1999)MATHCrossRefGoogle Scholar
  6. 6.
    Il’inskii, A.: A probabilistic approach to q-polynomial coefficients, Euler and Stirling numbers I. Mat. Fiz. Anal. Geom. 11(4), 434–448 (2004)MATHMathSciNetGoogle Scholar
  7. 7.
    Il’inskii, A., Ostrovska, S.: Convergence of generalized Bernstein polynomials. J. Approx. Theory 116, 100–112 (2002)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Lorentz, G.G.: Bernstein Polynomials. Chelsea, New York (1986)MATHGoogle Scholar
  9. 9.
    Lupaş, A.: A q-analogue of the Bernstein operator. University of Cluj-Napoca, Seminar on numerical and statistical calculus, No. 9 (1987)Google Scholar
  10. 10.
    Nevanlinna, R.: Analytic Functions. Springer, Berlin Heidelberg New York (1970)MATHGoogle Scholar
  11. 11.
    Oruç, H., Tuncer, N.: On the convergence and iterates of q-Bernstein polynomials. J. Approx. Theory 117, 301–313 (2002)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Ostrovska, S.: q-Bernstein polynomials and their iterates. J. Approx. Theory 123, 232–255 (2003)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Ostrovska, S.: On the q-Bernstein polynomials. Adv. Stud. Contemp. Math. 11(2), 193–204 (2005)MATHMathSciNetGoogle Scholar
  14. 14.
    Ostrovska, S.: On the improvement of analytic properties under the limit q-Bernstein operator. J. Approx. Theory 138, 37–53 (2006)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Ostrovska, S.: The approximation by q-Bernstein polynomials in the case q↓1. Arch. Math. 86(3), 282–288 (2006)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Phillips, G.M.: On generalized Bernstein polynomials. In: Griffits, D.F., Watson, G.A. (eds.) Numerical Analysis: A.R. Mitchell 75th Birthday, volume, pp. 263–269. World Science, Singapore (1996)Google Scholar
  17. 17.
    Phillips, G.M.: Bernstein polynomials based on the q-integers. Ann. Numer. Math. 4, 511–518 (1997)MATHMathSciNetGoogle Scholar
  18. 18.
    Phillips, G.M.: Interpolation and Approximation by Polynomials. Springer, Berlin Heidelberg New York (2003)MATHGoogle Scholar
  19. 19.
    Titchmarsh, E.C.: Theory of Functions. Oxford University Press, London (1986)Google Scholar
  20. 20.
    Videnskii, V.S.: On some classes of q-parametric positive operators. Oper. Theory Adv. Appl. 158, 213–222 (2005)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Videnskii, V.S.: On the polynomials with respect to the generalized Bernstein basis. In: Problems of Modern Mathematics and Mathematical Education, Hertzen Readings, pp. 130–134. St.-Petersburg, Russia (2005) (Russian) Google Scholar
  22. 22.
    Wang, H.: Korovkin-type theorem and application. J. Approx. Theory 132(2), 258–264 (2005)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Wang, H., Meng, F.: The rate of convergence of q-Bernstein polynomials for 0 < q < 1. J. Approx. Theory 136(2), 151–158 (2005)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Wang, H.: Voronovskaya type formulas and saturation of convergence for q-Bernstein polynomials for 0 < q < 1. J. Approx. Theory 145(2), 182–195 (2007)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Department of MathematicsAtilim UniversityIncekTurkey

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