İki Boyutlu Bernstein-Stancu Operatörlerinin Bir Genellemesi

Abstract

Çalışmamızın amacı, belirli bir aralıkta tanımlı iki boyutlu (p, q)-Bernstein-Stancu operatörlerinin bir genellemesini vermektir. Ayrıca, bu operatörlerin bazı direkt sonuçları oluşturularak, Lipschitz tipi fonksiyonlar ve süreklilik modülü ile yaklaşım hızı incelenmiştir.

Keywords

• (p,q)-tamsayı
• Korovkin tipi yaklaşım
• (p,q)-Bernstein Stancu operatörleri

A Generalization of Two Dimensional Bernstein-Stancu Operators

Abstract

The aim in our study is giving a generalization of the two-dimensional (p,q)-Bernstein-Stancu operators in a particular domain. In addition, by creating some direct results of these operators, rate of convergence is studied by Lipschitz type functions and modulus of continuity.

Keywords

• (p,q)-integer
• Korovkin type approximation
• (p,q)-Bernstein Stancu operators

References

1. Bernstein, S. (1912). Démostration du théoréme de Weierstrass fondée sur le calcul de probabilités. Communications de la Soci´et´e math´ematique de Kharkow, 13(1), 1-2.
2. Karaisa, A. (2016). On the approximation properties of bivariate (p,q)-Bernstein operators, arXiv:1601.05250.
3. Mursaleen, M., Ansari, K.J., & Khan, A. (2015). On (p,q)-analogue of Bernstein operators. Applied Mathematics and Computation, 266, 874-882. https://doi.org/10.1016/j.amc.2015.04.090 (Erratum to “On (p,q)-analogue of Bernstein Operators” Applied Mathematics and Computation, 278, 70-71. https://doi.org/10.1016/j.amc.2016.02.008).
4. Mursaleen, M., Ansari, K.J., & Khan, A. (2016). Some approximation results by (p,q)-analogue of Bernstein-Stancu operators, Applied Mathematics and Computation, 264, 392-402. https://doi.org/10.1016/j.amc.2015.03.135
5. Mursaleen, M., Nasiruzzaman, M., & Nurgali, A. (2015). Some approximation results on Bernstein-Schurer operators defined by (p,q)-integers., Journal of Inequalities and Applications, 249, https://doi.org/10.1186/s13660-015-0767-4
6. Kang, S. M., Rafiq, A., Acu, A. M., Ali, F., & Kwun, Y. C. (2016). Some approximation properties of (p,q)-Bernstein operators. Journal of Inequalities and Applications, 169, 1-10. https://doi.org/10.1186/s13660-016-1111-3
7. Khan, A., & Sharma, V. (2018). Statistical Approximation by (p,q)-analogue of Bernstein-Stancu operators. Azerbaijan Journal of Mathematics, 8(2), 100-121.
8. Ansari, K. J., & Karaisa, A. (2017). On the approximation by Chlodowsky type generalization of (p,q)-Bernstein operators. International Journal of Nonlinear Analysis and Applications, 8, 181-200. https://doi:10.22075/ijnaa.2017.1827.1479

9. Acar, T., Aral, A., & Mohiuddine, S. A. (2018). Approximation by bivariate (p,q)-Bernstein-Kantorovich operators. Iranian Journal of Science and Technology, Transactions A: Science, 42, 655–662. https://doi.org/10.1007/s40995-016-0045-4
10. Karahan, D., & Izgi, A. (2018). On approximation properties of (p,q)-Bernstein operators. European Journal of Pure and Applied Mathematics, 11(2), 457-467. https://doi.org/10. 29020/nybg.ejpam.v11i2.3213
11. Cevik, E. (2019). Approximation properties of modified (p,q)-Bernstein type operators. (Thesis no. 562017), [M.S. thesis, Harran University].
12. Barbosu, D. (2000). Some generalized bivariate Bernstein operators. Miskolc Mathematical Notes, 1(1), 3-10.
13. Gonul Bilgin, N., & Cetinkaya M. (2018). Approximation by three-dimensional q-Bernstein-Chlodowsky polynomials. Sakarya University Journal of Science, 22 (6), 1774-1786. https://doi: 10.16984/saufenbilder.348912
14. Karahan, D., & Izgi, A. (2018). Approximation properties of Bernstein-Kantorovich type operators of two variables. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68 (2), 2313-2323. https://doi: 10.31801/cfsuasmas.558169
15. Srivastava, H. M., Ansari, K. J., Ozger, F. & Odemis Ozger, Z. (2021). A Link between Approximation Theory and Summability Methods via Four-Dimensional Infinite Matrices. Mathematics, 9, 1895. https://doi.org/10.3390/math9161895
16. Anastassiou, G. A., & Gal, S. G. (2000). Approximation Theory: Moduli of continuity and global smoothness preservation. Springer Science & Business Media, LLC.