- Open Access
© Özarslan and Vedi; licensee Springer. 2013
- Received: 19 November 2012
- Accepted: 9 July 2013
- Published: 1 October 2013
In the present paper, we introduce the q-Bernstein-Schurer-Kantorovich operators. We give the Korovkin-type approximation theorem and obtain the rate of convergence of this approximation by means of the first and the second modulus of continuity. Moreover, we compute the order of convergence of the operators in terms of the elements of Lipschitz class functions and the modulus of continuity of the derivative of the function.
MSC: 41A10, 41A25, 41A36.
- q-integral operator
- positive linear operators
- q-Bernstein operators
- modulus of continuity
In 1987, q-based Bernstein operators were defined and studied by Lupaş . In 1996, another q-based Bernstein operator was proposed by Phillips . Then the q-based operators have become an active research area (see [5–9] and ).
where . If we choose in (1.1), we get the classical q-Bernstein operators .
Afterwards, several properties and results of the operators defined by (1.1), such as the order of convergence of these operators by means of Lipschitz class functions, the first and the second modulus of continuity and the rate of convergence of the approximation process in terms of the first modulus of continuity of the derivative of the function, were given by the authors . On the other hand, q-Szasz-Schurer operators were discussed in .
Then she obtained the first three moments and gave the rate of convergence of the approximation process in terms of the first modulus of continuity .
So, throughout this paper, we will use the following results, which are computed directly by the tools of q-calculus.
Recall that the first three moments of the q-Bernstein-Schurer operators were given by Muraru in  as follows.
We organize the paper as follows.
Firstly, in section two, we define the q-Bernstein-Schurer-Kantorovich operators and obtain the moments of them. In section three, we obtain the rate of convergence of the q-Bernstein-Schurer-Kantorovich operators in terms of the first modulus of continuity. Also we give the order of approximation by means of Lipschitz class functions and the first and the second modulus of continuity. Furthermore, we compute the degree of convergence of the approximation process in terms of the first modulus of continuity of the derivative of the function.
for any real number , and . It is clear that is a linear and positive operator for .
For the first three moments and the first and the second central moment, we state the following lemma.
- (ii)Using (1.1), (1.4) and Lemma 1.1, we have
- (iii)From (1.1), (1.4), (1.5) and then Lemma 1.1, we can calculate the as follows:
- (iv)It is obvious that
- (v)Direct calculations yield,(2.2)
By Korovkin’s theorem, we can state the following theorem. □
provided that with and that .
In this section, we compute the rate of convergence of the operators in terms of the modulus of continuity, elements of Lipschitz classes and the first and the second modulus of continuity of the function. Furthermore, we calculate the rate of convergence in terms of the first modulus of continuity of the derivative of the function.
where is the modulus of continuity of f and , which is given as Lemma 2.1.
The proof is concluded. □
where is the same as in Theorem 3.1.
Hence, the desired result is obtained. □
Hence we get the result. □
Now, we compute the rate of convergence of the operators in terms of the modulus of continuity of the derivative of the function.
This completes the proof. □
In this paper, we obtain many results in the pointwise sense. On the other hand, we see that the interval is bounded and closed, and also f is continuous on it, so these results can be given in the uniform sense.
- Schurer, F: Linear Positive Operators in Approximation Theory. Math. Inst., Techn. Univ. Delf Report (1962)Google Scholar
- Barbosu D: A survey on the approximation properties of Schurer-Stancu operators. Carpath. J. Math. 2004, 20: 1–5.MATHMathSciNetGoogle Scholar
- Lupaş AA: q -Analogue of the Bernstein operators. 9. Seminar on Numerical and Statistical Calculus, University of Cluj-Napoca 1987, 85–92.Google Scholar
- Phillips GM: On generalized bernstein polynomials. 98. In Numerical Analysis. World Scientific, River Edge; 1996:263–269.View ArticleGoogle Scholar
- Büyükyazıcı İ, Sharma H: Approximation properties of two-dimensional q -Bernstein-Chlodowsky-Durrmeyer operators. Numer. Funct. Anal. Optim. 2012, 33(2):1351–1371.MATHMathSciNetGoogle Scholar
- Büyükyazıcı İ, Atakurt Ç: On Stancu type generalization of q -Baskakov operators. Math. Comput. Model. 2010, 52(5–6):752–759. 10.1016/j.mcm.2010.05.004MATHView ArticleGoogle Scholar
- Gupta V, Finta Z: On certain q -Durrmeyer type operators. Appl. Math. Comput. 2009, 209(2):415–420. 10.1016/j.amc.2008.12.071MATHMathSciNetView ArticleGoogle Scholar
- Mahmudov NI, Sabancıgil P: q -Parametric Bleimann Butzer and Hahn operators. J. Inequal. Appl. 2008., 2008: Article ID 816367Google Scholar
- Ostrovska S: q -Bernstein polynomials and their iterates. J. Approx. Theory 2003, 123: 232–255. 10.1016/S0021-9045(03)00104-7MATHMathSciNetView ArticleGoogle Scholar
- Wang H, Wu XZ: Saturation of convergence for q -Bernstein polynomials in the case . J. Math. Anal. Appl. 2008, 337: 744–750. 10.1016/j.jmaa.2007.04.014MATHMathSciNetView ArticleGoogle Scholar
- Muraru CV: Note on q -Bernstein-Schurer operators. Stud. Univ. Babeş–Bolyai, Math. 2011, 56: 489–495.MathSciNetGoogle Scholar
- Kac V, Cheung P: Quantum Calculus. Springer, New York; 2002.MATHView ArticleGoogle Scholar
- Vedi T, Özarslan MA: Some properties of q -Bernstein-Schurer operators. J. Appl. Funct. Anal. 2013, 8(1):45–53.MATHMathSciNetGoogle Scholar
- Özarslan MA: q -Szasz Schurer operators. Miskolc Math. Notes 2011, 12: 225–235.MATHMathSciNetGoogle Scholar
- Duman O, Özarslan MA, Doğru O: On integral type generalizations of positive linear operators. Stud. Math. 2006, 176(1):1–12. 10.4064/sm176-1-1View ArticleGoogle Scholar
- Duman O, Özarslan MA, Vecchia BD: Modified Szasz-Mirakjan-Kantorovich operators preserving linear functions. Turk. J. Math. 2009, 33(2):151–158.MATHGoogle Scholar
- Özarslan MA, Duman O: Global approximation properties of modified SMK operators. Filomat 2010, 24(1):47–61. 10.2298/FIL1001047OMATHMathSciNetView ArticleGoogle Scholar
- Özarslan MA, Duman O: Local approximation behavior of modified SMK operators. Miskolc Math. Notes 2010, 11(1):87–99.MATHMathSciNetGoogle Scholar
- Özarslan MA, Duman O, Srivastava HM: Statistical approximation results for Kantorovich-type operators involving some special functions. Math. Comput. Model. 2008, 48(3–4):388–401. 10.1016/j.mcm.2007.08.015MATHView ArticleGoogle Scholar
- Dalmanoğlu Ö: Approximation by Kantorovich type q -Bernstein operators. MATH’07: Proceedings of the 12th WSEAS Intenational Conference on Applied Mathematics Egypt, 2007, 29–31.Google Scholar
- DeVore RA, Lorentz GG: Constructive Approximation. Springer, Berlin; 1993.MATHView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.