On Bivariate Complex Schurer-type Stancu Operators

Abstract

The study focuses on the approximation features of the bivariate generalization of the complex Schurer form of Stancu-type operators. We have obtained a Voronovskaja type solution that provides quantitative estimates for bivariate complex operators coupled to analytic functions. Furthermore, the exact order of approximation is provided.

Keywords

• bivariate complex opertators
• Schurer-type Stancu operators
• rate of convergence
• exact order.

References

1. Acar, T., Aral, A., Approximation properties of two dimensional Bernstein-Stancu-Chlodowsky operators, Le Matematiche 68(2)(2013), 15–31.
2. Bozkurt, K., Özsaraç, F., Aral, A., Bivariate Bernstein polynomials that reproduce exponential functions, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat.,70(1)(2021), 541–554.
3. Çetin, N., A new generalization of complex Stancu operators, Mathematical Methods in the Applied Sciences, 42(2019), 5582–5594.
4. Çetin, N., Başcanbaz-Tunca, G., Approximation by a new complex generalized Bernstein operator, An. Univ. Oradea Fasc. Mat. 26(2)(2019), 129–141.
5. Çetin, N., A new complex generalized Bernstein-Schurer operator, Carpathian J. Math., 37(1)(2021), 81–89.
6. Çetin, N., Manav Mutlu, N., Complex generalized Stancu-Schurer operators, Mathematica Slovaca, 74(5)(2024), 1215–1232.
7. Gal, S.G., Approximation by Complex Bernstein and Convoluation Type Operators, World Scientific, Sigapore, 2009.
8. Gal, S.G., Approximation by Complex Bernstein and Convolution Type Operators, Series on Concrete and Applicable Mathematics, 8. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, USA, 2009.

9. Manav Mutlu, N., On new bivariate Schurer-Stancu-type operators, (submitted).
10. Özden, D.S., Arı, D.A., Approximation by a complex q-Baskakov-Stancu operator in compact disks, Journal of Inequalities and Applications, (2014), 1–15.
11. Paltanea, R., Durrmeyer type operators on a simplex, Constructive Mathematical Analysis, 4(2)(2021), 215–228.
12. Stancu, D.D., Approximation of functions by means of a new generalized Bernstein operator, Calcolo, 20(1983), 211–229.
13. Stancu, D.D., Quadrature formulas constructed by using certain linear positive operators, Numerical Integration (Proc. Conf., Oberwolfach,1981), 57(1982), 241–251.
14. Stancu, D.D., Approximation of functions by means of a new generalized Bernstein operator, Calcolo, 20(1983) 211–229.
15. Yang, R., Xiong, J., Cao, F., Multivariate Stancu operators defined on a simplex, Appl. Math. Comput., 138(2003), 189–198.