Approximation properties of the univariate and bivariate Bernstein-Stancu operators of max-product kind

Abstract

This paper presents the nonlinear maximum product type of univariate and bivariate Bernstein–Stancu operators and uses new definitions to investigate the approximation properties. The order of approximation obtained with the nonlinear maximum product type of operator sequences would be better than the degree of approximation of the known linear operator sequences.

Keywords

• Bernstein-Stancu operators
• nonlinear operators
• degree of approximation

References

1. Altomare, F., Campiti, M., Korovkin-Type Approximation Theory and Its Applications, Walter de Gruyter, Berlin, 1994.
2. Korovkin, P. P., Linear Operators and Approximation Theory, Hindustan Publ. Corp., India, 1960.
3. Stancu, D. D., Asupra unei generaliz˘ari a polinoamelor lui Bernstein, Studia Universitatis Babeş-Bolyai, 14(2) (1969), 31-45 (in Romanian).
4. Bede, B., Coroianu, L., Gal, S. G., Approximation and shape preserving properties of the Bernstein operator of max-product kind, Intern, J. Math. and Math. Sci., 26 pages (2009). doi:10.1155/2009/590589
5. Bede, B., Gal, S. G., Approximation by nonlinear Bernstein and Favard-Szasz-Mirakjan operators of max-product kind, Journal of Concrete and Applicable Mathematics, 8(2) (2010), 193-207.
6. Bede, B., Coroianu, L., Gal, S. G., Approximation by Max-Product Type Operators, Heidelberg, Springer, 2016.
7. Coroianu, L., Gal, S. G,. Approximation by nonlinear generalized sampling operators of max-product kind, Sampl. Theory Signal Image Process, 9 (2010), 59-75. https://doi.org/10.1007/BF03549524
8. Coroianu, L., Gal, S. G., Approximation by max-product sampling operators based on sinc-type kernels, Sampl. Theory Signal Image Process, 10 (2011), 211-230. https://doi.org/10.1007/BF03549542

9. Hildebrandt, T. H., Schoenberg, I. J., On linear functional operations and the moment problem, Ann. Math., 34(2) (1933), 317-328.
10. Butzer, P. L., On two-dimensional Bernstein polynomials, Can. J. Math., 5 (1953), 107-113.
11. Martinez, F. L., Some properties of two-dimensional Bernstein polynomials, Journal of approximation theory, 59(3) (1989), 300-306. https://doi.org/10.1016/0021-9045(89)90095-6
12. Kırcı Serenbay, S., Yavuz, H., Approximation Of Modified Bernstein-Stancu Operators Of Maximum-Product Type, presented at the İzdaş Kongre, Ankara, Turkey, 2021.
13. Acar, E., Kırcı Serenbay, S., Approximation by nonlinear q-Bernstein-Chlodowsky operators, TWMS J. App. and Eng. Math., 14(1) (2024), 42-51.
14. Acar, E., Özalp Guller, Ö., Kırcı Serenbay, S., Approximation by nonlinear Meyer-König and Zeller operators based on q-integers, International Journal of Mathematics and Computer in Engineering, 2(2) (2024), 71-82.
15. Acar, E., Kırcı Serenbay, S., Özalp Guller, Ö., Approximation by nonlinear Bernstein-Chlodowsky operators of Kantorovich type, Filomat, 37(14) (2023), 4621-4627. https://doi.org/10.2298/FIL2314621A
16. Özalp Guller, Ö., Acar, E., Kırcı Serenbay, S., Nonlinear bivariate Bernstein-Chlodowsky operators of maximum product type, Journal of Mathematics, (2022). https://doi.org/10.1155/2022/4742433
17. Acar, E., Holhoş, A., Kırcı Serenbay, S., Polynomial weighted approximation by Szasz-Mirakyan operators of max-product type, Kragujevac Journal of Mathematics, 49(3) (2022), 365-373. 10.46793/KgJMat2503.365A
18. Gairola, A. R., Singh, A., Rathour, L., Mishra, V. N., Improved rate of approximation by modification of Baskakov operator, Operators and Matrices, 16(4), (2022), 1097-1123. dx.doi.org/10.7153/oam-2022-16-72
19. Gairola, A. R., Maindola, S., Rathour, L., Mishra, L. N., Mishra, V. N., Better uniform approximation by new Bivariate Bernstein Operators, International Journal of Analysis and Applications, 20(60) (2022), 1-19. https://doi.org/10.28924/2291-8639-20-2022-60
20. Gairola, A. R., Bisht, N., Rathour, L., Mishra, L. N., Mishra, V. N., Order of approximation by a new univariate Kantorovich Type Operator, International Journal of Analysis and Applications, 21 (2023), 1-17. https://doi.org/10.28924/2291-8639-21-2023-106
21. Mishra, V. N., Khatri, K., Mishra, L. N., Deepmala, Inverse result in simultaneous approximation by Baskakov-Durrmeyer-Stancu operators, Journal of Inequalities and Applications, 586 (2013). https://doi.org/10.1186/1029-242X-2013-586
22. Yeşilnacar Binmar, A. K., Aproximation properties two biviariate maximum product type operators, Master Thesis, Harran University, Şanlıurfa, Türkiye, 2023.