Yet another Kantorovich variant of Lupaş-Jain operators

Abstract

This paper proposes a novel Kantorovich-type extension of the Lupaş-Jain operators, aiming to enhance their approximation capabilities. The investigation begins by establishing weighted uniform approximation properties for the newly defined operators. Following this, a rigorous quantitative estimate is derived. Then, we examine the order of approximation by employing a modulus of continuity, the Lipschitz class and the Lipschitz-type maximal function. To conclude, numerical examples are presented to substantiate the theoretical findings.

Keywords

• Lupaş-Jain operators
• Kantorovich operators
• Weighted approximation

Lupaş-Jain operatörlerinin bir başka Kantorovich genelleştirmesi

Abstract

Bu makale, Lupaş-Jain operatörlerinin yaklaşım özelliklerini geliştirmek amacıyla, Kantorovich tipi yeni bir genelleştirme önermektedir. Araştırma, yeni tanımlanan operatörler için ağırlıklı düzgün yaklaşım özelliklerinin belirlenmesiyle başlamaktadır. Ardından, bir nicel tahmin türetilmektedir. Daha sonra, süreklilik modülü, Lipschitz sınıfı ve Lipschitz tipi maksimum fonksiyon kullanılarak yaklaşımın derecesi incelenmektedir. Sonuç olarak, teorik bulguları doğrulamak için sayısal örnekler sunulmaktadır.

Keywords

• Lupaş-Jain operatörleri
• Kantorovich operatörleri
• Ağırlıklı yaklaşım

References

1. [1] Başcanbaz-Tunca, G., Bodur, M., and Söylemez, D., “On Lupaş-Jain operators”, Studia Universitatis Babeş-Bolyai Mathematica, 63(4):525-537, (2018). DOI: https://doi.org/10.24193/subbmath.2018.4.08
2. [2] Lupaş, A., “The approximation by some positive linear operators”, In: Proceedings of the International Dortmund Meeting on Approximation Theory (M.W. Muller et al., eds.), Akademie Verlag, Berlin 201-229, (1995).
3. [3] Jain, G.C., “Approximation of functions by a new class of linear operators”, Journal of the Australian Mathematical Society, 13(3):271-276, (1972). DOI: https://doi.org/10.1017/S1446788700013689
4. [4] Gasper, G., Rahman, M., “Basic hypergeometric series”, Cambridge University Press, (2004).
5. [5] Patel, P., and Mishra, V.N., “On new class of linear and positive operators”, Bollettino dell'Unione Matematica Italiana, 8:81-96, (2015). DOI: https://doi.org/10.1007/s40574-015-0026-0
6. [6] Agratini, O., “A stop over Jain operators and their generalizations”, Analele Universității de Vest din Timișoara Seria Matematică-Informatică, 2:28-42, (2018). DOI: https://doi.org/10.2478/awutm-2018-0014
7. [7] Bodur, M., “Modified Lupaş-Jain operators”, Mathematica Slovaca, 70(2):431-440, (2020). DOI: https://doi.org/10.1515/ms-2017-0361
8. [8] Patel, P., “Some approximation results of Kantorovich type operators”, Journal of Computational Analysis and Applications, 29(1):52-67, (2021).

9. [9] Gupta, V., Muraru, C.V.P., and Radu, V.A., “Convergence of certain hybrid operators”, Rocky Mountain Journal of Mathematics, 51:1249-1258, (2021). DOI: https://doi.org/10.1216/rmj.2021.51.1249
10. [10] Patel, P., and Bodur, M., “On integral generalization of Lupaş-Jain operators”, Filomat, 36(3):729-740, (2022). DOI: https://doi.org/10.2298/fil2203729p
11. [11] Patel, P., Rajawat, R.S., Rathour, L., and Mishra, V.N., “Statistical convergence of Lupaş-Jain operators”, In AIP Conference Proceedings, 3005(1), AIP Publishing, (2024). DOI: https://doi.org/10.1063/5.0210525
12. [12] Gülfırat, M., and Bodur, M., “Weighted approximation results for Lupaş-Jain operators via summability methods”, Matematički Vesnik, 78(1):35-43, (2026). DOI: https://doi.org/10.57016/MV-HRZW3917
13. [13] Altomare, F., Cappelletti Montano, M., Leonessa V., and Raşa, I., “A generalization of Kantorovich operators for convex compact subsets”, Banach Journal of Mathematical Analysis, 11(3):591-614, (2017). DOI: https://doi.org/10.1215/17358787-2017-0008
14. [14] Kantorovich, L.V., “Sur certains développements suivant les polynômes de la forme de S. Bernstein”, I, II, Comptes Rendus de l'Académie des Sciences de l'URSS, 563-568, 595-600, (1930).
15. [15] Gadzhiev, A.D., “Theorems of Korovkin type”, Mathematical Notes of the Academy of Sciences of the USSR 20, 995-998 (1976). DOI: https://doi.org/10.1007/BF01146928
16. [16] Holhoş, A., “Quantitative estimates for positive linear operators in weighted spaces”, General Mathematics, 16:99-110, (2008).
17. [17] DeVore, R.A., and Lorentz, G.G., “Constructive approximation”, series Grundlehren der Mathematischen Wissenschaften. Springer Verlag, Berlin, 303, 1993.
18. [18] Özarslan, M.A., and Aktuğlu, H., “Local approximation properties for certain King type operators”, Filomat, 27(1):173--181, (2013). DOI: https://doi.org/10.2298/FIL1301173O
19. [19] Lenze, B., “On Lipschitz-type maximal functions and their smoothness spaces”, Nederlandse Akademie van Wetenschappen. Proceedings. Series A. Indagationes Mathematicae, 50(1):53-63, (1988). DOI: https://doi.org/10.1016/1385-7258(88)90007-8