The rate of pointwise approximation of positive linear operators based on q-integer

  • G. Nowak
  • V. Gupta
Article

The paper deals with positive linear operators based on q-integer. The rate of convergence of these operators is established. For these operators, we present Voronovskaya-type theorems and apply them to q-Bernstein polynomials and q-Stancu operators.

Keywords

Approximation Property Pointwise Convergence Bernstein Polynomial POINTWISE Approximation Positive Linear Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    E. W. Cheney, Introduction to Approximation Theory, Chelsea, New York (1982).MATHGoogle Scholar
  2. 2.
    T. Ernst, “A new notation for q-calculus and a new q-Taylor formula,” Uppsala Univ., Rep. Depart. Math., 62–64 (1999).Google Scholar
  3. 3.
    Z. Finta, “Direct and converse results for Stancu operator,” Period. Math. Hung., 44, No. 1, 1–16 (2002).MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Z. Finta, “On approximation properties of Stancu’s operators,” Stud. Univ. BABE-SBOLYAI. Math., 47, No. 4, 47–55 (2002).MathSciNetGoogle Scholar
  5. 5.
    H. H. Gonska and J. Meier, “Quantitative theorems on approximation by Bernstein–Stancu operators,” Calcolo, 21, No. 4, 317–335 (1984).MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    T. N. T. Goodman, H. Oruc, and G. M. Phillips, “Convexity and generalized Bernstein polynomials,” Proc. Edinburgh Math. Soc., 42, 179–190 (1999).MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    J. Hoschek and D. Lasser, Fundamentals of Computer Aided Geometric Design, Peters, Wellesley, MA (1993).MATHGoogle Scholar
  8. 8.
    A. II’inskii and S. Ostrovska, “Convergence of generalized Bernstein polynomials,” J. Approx. Theory, 116, No. 1, 100–112 (2002).MathSciNetCrossRefGoogle Scholar
  9. 9.
    F. H. Jackson, “On q-definite integrals,” Quart. J. Pure Appl. Math., 41, 193–203 (1910).MATHGoogle Scholar
  10. 10.
    V. Kac and P. Cheung, Quantum Calculus, Springer, New York (2002).MATHCrossRefGoogle Scholar
  11. 11.
    T. Kim, “q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients,” Russ. J. Math. Phys., 15, 51–57 (2008).MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    G. G. Lorentz, Bernstein Polynomials, University of Toronto Press, Toronto (1953).MATHGoogle Scholar
  13. 13.
    G. Nowak, “Approximation properties for generalized q-Bernstein polynomials,” J. Math. Anal. Appl., 350, No. 1, 50–55 (2009).MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    G. Nowak, “A de Casteljau algorithm for q-Bernstein–Stancu polynomials,” Abstr. Appl. Anal., 211, Article ID 609431 (2011).Google Scholar
  15. 15.
    G. M. Phillips, “Bernstein polynomials based on the q-integers,” Ann. Numer. Math., 4, 511–518 (1997).MathSciNetMATHGoogle Scholar
  16. 16.
    G. M. Phillips, “A de Casteljau algorithm for generalized Bernstein polynomials,” BIT, 1996, 36, 232–236.Google Scholar
  17. 17.
    H. Oruc and G. M. Phillips, “A generalization of the Bernstein polynomials,” Proc. Edinburgh Math. Soc., 42, 403–413 (1999).MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    H. Oruc and G. M. Phillips, “Explicit factorization of the Vandermonde matrix,” Linear Algebra Its Appl., 315, 113–123 (2000).MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    S. Ostrovska, “q-Bernstein polynomials and their iterates,” J. Approx. Theory, 123, No. 2, 232–255 (2003).MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    S. Ostrovska, “On the limit q-Bernstein operators,” Math. Balkan, 18, 165–172 (2004).MathSciNetMATHGoogle Scholar
  21. 21.
    S. Ostrovska, “On the improvement of analytic properties under the limit q-Bernstein operators,” J. Approx. Theory, 138, No. 1, 37–53 (2006).MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    P. Pych-Taberska, “Some approximation properties of Bernstein and Kantorovic polynomials,” Funct. Approx., 6, 57–67 (1978).MathSciNetMATHGoogle Scholar
  23. 23.
    P. Pych-Taberska, “On the rate of pointwise convergence of Bernstein and Kantorovic polynomials,” Funct. Approx., 16, 63–76 (1988).MathSciNetGoogle Scholar
  24. 24.
    P. Pych-Taberska, “Rate of pointwise convergence of Bernstein polynomials for some absolutely continuous functions,” J. Math. Anal. Appl., 212, 9–19 (1997).MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    P. M. Rajkovic, M. S. Stankovic, and S. D. Marinkovic, “Mean value theorems in q-calculus,” Math. Vesnic, 54, 171–178 (2002).MathSciNetMATHGoogle Scholar
  26. 26.
    D. D. Stancu, “Approximation of functions by a new class of linear polynomial operators,” Rev. Roum. Math. Pures Appl., 13, No. 8, 1173–1194 (1968).MathSciNetMATHGoogle Scholar
  27. 27.
    V. S. Videnskii, “On some classes of q-parametric positive operators,” Operator Theory: Adv. Appl., 158, 213–222 (2005).MathSciNetCrossRefGoogle Scholar
  28. 28.
    H. Wang and F. Meng, “The rate of convergence of q-Bernstein polynomials for 0 < q < 1;” J. Approx. Theory, 136, No. 2, 151–158 (2005).MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    H. Wang, “Korovkin-type theorem and application,” J. Approx. Theory, 132, No. 2, 258–264 (2005).MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    H.Wang, “Voronovskaya-type formulas and saturation of convergence for q-Bernstein polynomials for 0 < q < 1;” J. Approx. Theory, 145, 182–195 (2007).MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  • G. Nowak
    • 1
  • V. Gupta
    • 2
  1. 1.Sroda WielkopolskaPoland
  2. 2.New DelhiIndia

Personalised recommendations