Results on Bivariate Modified (p, q)-Bernstein Type Operators

Year 2023, Volume: 36 Issue: 2 , 845 - 860 , 01.06.2023

Nazmiye Gönül Bilgin , Melis Eren

https://doi.org/10.35378/gujs.868893

Cited By: 2

https://izlik.org/JA68EM64NH

Abstract

Here, we construct a modification of the (𝑝,𝑞)-Bernstein operators for the two-dimensional case. We study some important properties of these new operators. We estimate the rate of convergence of these operators using modulus of continuity then we give these estimation for functions belonging to class 𝐿𝑖𝑝𝑀(𝛼1,𝛼2).

Keywords

(p,q)-Bernstein operators , (p,q)- integer , Modulus of continuity

References

• [1] Bernstein, S.N., “Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités”, Communications of the Kharkov Mathematical Society, 13(2): 1-2, (1912).
• [2] Phillips, G.M., “Bernstein polynomials based on the q-integers”, Annals of Numerical Mathematics, 4: 511-518, (1997).
• [3] Ostrovska, S., “q-Bernstein polynomials and their iterates”, Journal of Approximation Theory, 123(2): 232–255, (2003).
• [4] Wang, H., “Voronovskaya-type formulas and saturation of convergence for q-Bernstein polynomials for 0
• [5] Buyukyazici, I., “On the approximation properties of two-dimensional q-Bernstein-Chlodowsky polynomials”, Mathematical Communications, 14(2): 255-269, (2009).
• [6] Gonul Bilgin, N., and Cetinkaya, M., “Approximation by three-dimensional q-Bernstein-Chlodowsky polynomials”, Sakarya University Journal of Science, 22(6): 1774-1786, (2018).
• [7] Mursaleen, M., Ansari, J.A., and Khan, A., “On (p,q)-analogue of Bernstein operators”, Applied Mathematics and Computation, 278: 70–71, (2016).
• [8] Kanat, K., and Sofyalioglu, M., “Some approximation results for Stancu type Lupaş-Schurer operators based on (𝑝,)-integers”, Applied Mathematics and Computation, 317: 129-142, (2018).
• [9] Kanat, K., and Sofyalıoğlu, M., “Approximation by (p,q)-Lupaş–Schurer–Kantorovich operators”, Journal of Inequalities and Applications, 2018(1): 217-229, (2018).
• [10] Kanat, K., and Sofyalıoğlu, M., “On Stancu type generalization of (p,q)-Baskakov-Kantorovich operators”, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68(2): 1995-2013, (2019).
• [11] Cai, Q. B., and Zhou, G., “On (p, q)-analogue of Kantorovich type Bernstein–Stancu–Schurer operators”, Applied Mathematics and Computation, 276: 12-20, (2016).
• [12] Kanat, K., and Sofyalıoğlu, M., “Some approximation results for (p, q)-Lupaş-Schurer operators”, Filomat, 32(1), 217-229, (2018).
• [13] Acar, T., “(p, q)‐Generalization of Szász–Mirakyan operators”, Mathematical Methods in the Applied Sciences, 39(10): 2685-2695, (2016).
• [14] Kanat, K., and Sofyalıoğlu, M., “Approximation Properties of Stancu-Type (p,q)-Baskakov Operators”, Bitlis Eren University Journal of Science, 8(3): 889-902, (2019).
• [15] Bilgin N. G., and Eren M., “A Generalization of Two Dimensional Bernstein-Stancu Operators”, Sinop University Journal of Science, 6(2): 130-142, (2021).
• [16] Mishra V.N. and Pandey, S., “On (𝑝, 𝑞) Baskakov–Durrmeyer–Stancu operators”, Advances in Applied Clifford Algebras, 27: 1633-1646, (2017).
• [17] Acar, T., Aral, A. and Mohiuddine, S.A., “Approximation by bivariate (p,q)-Bernstein-Kantorovich operators”, Iranian Journal of Science and Technology, Transactions A: Science, 42: 655–662, (2018).
• [18] Karaisa, A., “On the approximation properties of bivariate (p,q)-Bernstein operators”, https://arxiv.org/abs/1601.05250v2, Access date: 28.01.2021
• [19] Izgi, A., and Karahan, D., “On approximation properties of generalised (p,q)-Bernstein operators”, European Journal of Pure And Applied Mathematics, 11(2): 457-467, (2018).
• [20] Cevik, E., “Approximation properties of modified (p,q)-Bernstein type operators”, MSc Thesis, Harran University Institute of Natural and Applied Sciences, Sanliurfa, 16-34, (2019).
• [21] Chakrabarti, R., and Jagannathan, R., “A (p,q)-oscillator realization of two-parameter quantum algebras”, Journal of Physics A: Mathematical and General, 24: L711-L718, (1991).
• [22] Cao, F., Ding, C., and Xu, Z., “On multivariate Baskakov operator”, Journal of Mathematical Analysis and Applications, 307(1): 274-291, (2005).