Statistical approximation to Chlodowsky type q-Bernstein-Schurer-Stancu-Kantorovich operators

Year 2017, Volume: 5 Issue: 1, 108 - 121, 30.04.2017

Behar Baxhaku , Fevzi Berisha

https://doi.org/10.36753/mathenot.421714

Cited By: 2

Abstract

In this paper we introduce two kinds of Chlodowsky-type q-Bernstein-Schurer-Stancu- Kantorovich
operators on the onbounded domain. The Korovkin type statistical approximation property of these
operators are investigated. We investigated the rate of convergence for this approximation by means of
the first and the second modulus of continuity. The rate of convergence is investigated by using Lipschitz
classes of functions and the modulus of continuity of the derivative of the function. Then, we obtain
point-wise estimate the Lipchitz type maximal function.

Keywords

Chlodowsky–type, q-Bernstein-Schurer-Stancu-Kantorivich, q-integer, modulus of continuity

References

• [1] Agratini, O., and Ogün D.. "Weighted approximation by q-Szász-King type operators." Taiwanese Journal of Mathematics (2010): 1283-1296.
• [2] Agrawal, P. N., Vijay Gupta, and A. Sathish Kumar. "On q-analogue of Bernstein–Schurer–Stancu operators." Applied Mathematics and Computation 219.14 (2013): 7754-7764.
• [3] Aral, A., V. Gupta, and R. P. Agarwal. Applications of q-calculus in operator theory. New York: Springer, 2013.
• [4] Area, I., et al. "Formulae relating little q-Jacobi, q-Hahn and q-Bernstein polynomials: application to q-Bézier curve evaluation." Integral Transforms and Special Functions 15.5 (2004): 375-385.
• [5] Dalmanoglu, Özge. "Approximation by Kantorovich type q-Bernstein operators." Proceedings of the 12th WSEAS International Conference on Applied Mathematics. World Scientific and Engineering Academy and Society (WSEAS), 2007.
• [6] Duman, O., and C. Orhan. "Statistical approximation by positive linear operators." Studia Mathematica 161.2 (2004): 187-197.
• [7] Freedman, Allen, and John Sember. "Densities and summability." Pacific Journal of Mathematics 95.2 (1981): 293-305.
• [8] Gadjiev, A. D., and C. Orhan. "Some approximation theorems via statistical convergence." Rocky Mountain J. Math 32.1 (2002).
• [9] Gal, S. G., and V. Gupta. "Approximation of vector-valued functions by q-Durrmeyer operators with applications to random and fuzzy approximation." Oradea Univ. Math. J 16 (2009): 233-242.
• [10] Gupta, Vijay. "Some approximation properties of q-Durrmeyer operators." Applied Mathematics and Computation 197.1 (2008): 172-178.
• [11] Gupta, Vijay, and Cristina Radu. "Statistical approximation properties of q-Baskakov-Kantorovich operators." Open Mathematics 7.4 (2009): 809-818.
• [12] Kac, V., and Pokman Ch.. Quantum calculus. Springer Science & Business Media, 2001.
• [13] Karsli, H., and Vijay G.. "Some approximation properties of q-Chlodowsky operators." Applied Mathematics and Computation 195.1 (2008): 220-229.
• [14] Lin, Qiu. "Statistical Approximation of-Bernstein-Schurer-Stancu-Kantorovich Operators." Journal of Applied Mathematics 2014 (2014).
• [15] Lupas¸ A. A q-analogue of the Bernstein operator. University of Cluj-Napoca, Seminar on numerical and statistical calculus, Nr. 9, (1987).
• [16] Ostrovska, S. "The sharpness of convergence results for q-Bernstein polynomials in the case q> 1." Czechoslovak Mathematical Journal 58.4 (2008): 1195-1206.
• [17] Ostrovska, Sofiya. "On the image of the limit q-Bernstein operator." Mathematical Methods in the Applied Sciences 32.15 (2009): 1964-1970.
• [18] Phillips, George M. "Bernstein polynomials based on the q-integers." Annals of Numerical Mathematics 4 (1996): 511-518.
• [19] Ren, Mei-Ying, and Xiao-Ming Zeng. "On statistical approximation properties of modified q-Bernstein-Schurer operators." Bulletin of the Korean Mathematical Society 50.4 (2013): 1145-1156.
• [20] Vedi, T., and Mehmet A. Ö. "Chlodowsky-type q-Bernstein-Stancu-Kantorovich operators." Journal of Inequalities and Applications 2015.1 (2015): 91.