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Journal of Function Spaces
Volume 2018, Article ID 6385451, 15 pages
https://doi.org/10.1155/2018/6385451
Research Article

On the Convergence of a Family of Chlodowsky Type Bernstein-Stancu-Schurer Operators

1School of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou 362000, China
2School of Applied Mathematics, Xiamen University of Technology, Xiamen 361024, China

Correspondence should be addressed to Qing-Bo Cai; moc.621@iacbq

Received 26 April 2018; Revised 15 June 2018; Accepted 27 June 2018; Published 18 July 2018

Academic Editor: Gestur Ólafsson

Copyright © 2018 Lian-Ta Shu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. S. N. Bernstein, “Démonstration du théorème de Weierstrass fondée sur la calcul des probabilités,” Comm. Soc. Math. Charkow Sér, vol. 13, pp. 1-2, 1912. View at Google Scholar
  2. S. A. Mohiuddine, T. Acar, and A. Alotaibi, “Construction of a new family of Bernstein-Kantorovich operators,” Mathematical Methods in the Applied Sciences, vol. 40, no. 3, pp. 7749–7759, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  3. T. Acar and A. Kajla, “Degree of Approximation for Bivariate Generalized Bernstein Type Operators,” Results in Mathematics, vol. 79, no. 2, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  4. S. A. Mohiuddine, T. Acar, and M. Alghamdi, “Genuine modified Bernstein-Durrmeyer operators,” Journal of Inequalities and Applications, vol. 104, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  5. Q.-B. Cai, B.-Y. Lian, and G. Zhou, “Approximation properties of λ-Bernstein operators,” Journal of Inequalities and Applications, vol. 61, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  6. A. Lupas, “A q-analogue of the Bernstein operator,” in Proceedings of the Seminar on Numerical and Statistical Calculus, pp. 85–92, University of Cluj-Napoca, 1987. View at MathSciNet
  7. G. M. Phillips, “Bernstein polynomials based on the -integers,” Annals of Numerical Mathematics, vol. 4, no. 1–4, pp. 511–518, 1997. View at Google Scholar · View at MathSciNet
  8. M. Mursaleen, K. J. Ansari, and A. Khan, “On -analogue of Bernstein operators,” Applied Mathematics and Computation, vol. 266, pp. 874–882, 2015. View at Publisher · View at Google Scholar · View at Scopus
  9. M. Mursaleen, K. J. Ansari, and A. Khan, “Some approximation results by (p, q)-analogue of Bernstein-Stancu operators,” Applied Mathematics and Computation, vol. 264, pp. 392–402, 2015. View at Publisher · View at Google Scholar
  10. S. M. Kang, A. Rafiq, A.-M. Acu, F. Ali, and Y. C. Kwun, “Some approximation properties of (p, q)-Bernstein operators,” Journal of Inequalities and Applications, vol. 169, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  11. A. Khan and V. Sharma, “Statistical approximation by (p, q)-analogue of Bernstein-Stancu operators,” 2017.
  12. T. Acar, A. Aral, and S. A. Mohiuddine, “Approximation by Bivariate (p, q)-Bernstein-Kantorovich Operators,” Iranian Journal of Science & Technology, vol. 42, no. 2, pp. 655–662, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  13. M. Mursaleen, F. Khan, and A. Khan, “Approximation by (p, q)-Lorentz polynomials on a compact disk,” Complex Analysis and Operator Theory, vol. 10, no. 8, pp. 1725–1740, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  14. M. Mursaleen, M. Nasiruzzaman, A. Khan, and K. J. Ansari, “Some approximation results on Bleimann-Butzer-Hahn operators defined by (p, q)-integers,” Filomat, vol. 30, no. 3, pp. 639–648, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  15. M. Mursaleen and A. Khan, “Generalized q-Bernstein-Schurer operators and some approximation theorems,” Journal of Function Spaces and Applications, vol. 2013, Article ID 719834, 7 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  16. U. Kadak, “On weighted statistical convergence based on (p, q)-integers and related approximation theorems for functions of two variables,” Journal of Mathematical Analysis and Applications, vol. 443, no. 2, pp. 752–764, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  17. T. Acar, P. N. Agrawal, and A. Sathish Kumar, “On a modification of (p, q)-Szász-Mirakyan operators,” Complex Analysis and Operator Theory, vol. 12, no. 1, pp. 1–3, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  18. T. Acar, A. Aral, and S. A. Mohiuddine, “On Kantorovich modification of (p, q)-Baskakov operators,” Journal of Inequalities and Applications, vol. 98, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  19. A. D. Gadjiev, R. O. Efendiev, and E. Ibikli, “Generalized Bernstein-Chlodowsky polynomials,” Rocky Mountain Journal of Mathematics, vol. 28, no. 4, pp. 1267–1277, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  20. E. A. Gadjieva and E. Ibikli, “Weighted approximation by Bernstein-Chlodowsky polynomials,” Indian Journal of Pure and Applied Mathematics, vol. 30, no. 1, pp. 83–87, 1999. View at Google Scholar · View at MathSciNet · View at Scopus
  21. H. Karsli and V. Gupta, “Some approximation properties of q-Chlodowsky operators,” Applied Mathematics and Computation, vol. 195, no. 1, pp. 220–229, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  22. A. E. Piriyeva, “On order of approximation of functions by generalized Bernstein-Chlodowsky polynomials,” Proceedings of Institute of Mathematics and Mechanics. National Academy of Sciences of Azerbaijan, vol. 21, pp. 157–164, 2004. View at Google Scholar · View at MathSciNet
  23. I. Büyükyazici and E. Ibikli, “Inverse theorems for Bernstein-Chlodowsky type polynomials,” Journal of Mathematics of Kyoto University, vol. 46, pp. 21–29, 2006. View at Google Scholar
  24. E. A. Gadjieva and T. K. Gasanova, “Approximation by two dimensional Bernstein-Chlodowsky polynomials in triangle with mobile boundary,” Transactions of Azerbaijan National Academy of Sciences. Series of Physical-Technical and Mathematical Sciences, vol. 20, no. 4, Math. Mech., pp. 47–51, 265, 2000. View at Google Scholar · View at MathSciNet
  25. E. Ibikli, “On approximation for functions of two variables on a triangular domain,” Rocky Mountain Journal of Mathematics, vol. 35, no. 5, pp. 1523–1531, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  26. V. N. Mishra, M. Mursaleen, S. Pandey, and A. Alotaibi, “Approximation properties of Chlodowsky variant of (p, q) Bernstein-Stancu-SCHurer operators,” Journal of Inequalities and Applications, vol. 176, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  27. M. N. Hounkonnou, J. Désiré, and B. Kyemba, “R(p, q)-calculus: differentiation and integration,” SUT Journal of Mathematics, vol. 49, pp. 145–167, 2013. View at Google Scholar
  28. R. Jagannathan and K. S. Rao, “Two-parameter quantum algebras, twin-basic numbers, and associated generalized hypergeometric series,” in Proceedings of the International Conference on Number Theory and Mathematical Physics, pp. 20-21, 2005.
  29. J. Katriel and M. Kibler, “Normal ordering for deformed boson operators and operator-valued deformed Stirling numbers,” Journal of Physics A: Mathematical and General, vol. 24, no. 9, pp. 2683–2691, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  30. P. N. Sadjang, “On the fundamental theorem of (p, q)-calculus and some (p, q)-Taylor formulas,” 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  31. V. Sahai and S. Yadav, “Representations of two parameter quantum algebras and p, q-special functions,” Journal of Mathematical Analysis and Applications, vol. 335, no. 1, pp. 268–279, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  32. Q.-B. Cai and X.-W. Xu, “A basic problem of (p, q)-Bernstein operators,” Journal of Inequalities and Applications, vol. 140, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  33. R. A. DeVore and G. G. Lorentz, Constructive Approximation, vol. 303, Springer, Berlin, Germany, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  34. H. G. Ilarslan and T. Acar, “Approximation by bivariate (p,q)-Baskakov–Kantorovich operators,” Georgian Mathematical Journal, 2016. View at Publisher · View at Google Scholar