A Generalization of Blending-Type Szasz–Mirakjan Operators and Their Approximation Properties

Year 2026, Volume: 14 Issue: 1 , 42 - 51 , 30.04.2026

Mohd Raiz , Hayatem Hamal , Khursheed Ansari

https://izlik.org/JA46WY59AW

Abstract

This paper introduces a new generalization of blending-type Szász-Mirakjan operators via an additional parameter $\alpha$. We investigate fundamental approximation properties including moment estimates, central moments, and local approximation results. Korovkin-type theorems are established to prove uniform convergence, while various tools are employed to study rates of convergence. Weighted approximation properties are examined in depth, analyzing the behavior of operators in weighted spaces and establishing convergence results for functions with polynomial growth. Furthermore, A-statistical approximation properties are thoroughly investigated, providing convergence results under weaker conditions than classical approaches. The theoretical findings are supported by comprehensive numerical and graphical analyses, demonstrating the effectiveness of the proposed operators. Error analysis confirms that approximation quality improves significantly as the parameter increases, with visual evidence showing uniform convergence behavior. Both global and local approximation properties are examined using moduli of smoothness and Peetre's $K$-functional in different function spaces. The results confirm that our operators provide enhanced approximation capabilities compared to existing ones.

Keywords

Blending-Type operators , Korovkin Theorem , Sz\ , Local approximation.

References

• [1] Sz´asz, O. 1950. “Generalization of S. Bernstein Polynomials to the Infinite Interval.” Journal of Research of the National Bureau of Standards 45: 239–245.
• [2] Alotaibi, A. 2023. “On the Approximation by Bivariate Sz´asz–Jakimovski–Leviatan-Type Operators of Unbounded Sequences of Positive Numbers.” Mathematics 16(4): 1009.
• [3] Alotaibi, A. 2022. “Approximation of GBS-Type q-Jakimovski–Leviatan–Beta Integral Operators in B¨ogel Space.” Mathematics 10(5): 675.
• [4] Cicek, H., and A. Izgi. 2022. “Approximation by Modified Bivariate Bernstein–Durrmeyer and GBS Bivariate Bernstein–Durrmeyer Operators on a Triangular Region.” Fundamental Journal of Mathematics and Applications 5: 135–144.
• [5] Izgi, A., and S. K. Serenbay. 2020. “Approximation by Complex Chlodowsky–Sz´asz–Durrmeyer Operators in Compact Disks.” Creative Mathematics and Informatics 29: 37–44.
• [6] Ayman-Mursaleen, M., M. Nasiruzzaman, N. Rao, M. Dilshad, and K. S. Nisar. 2024. “Approximation by the Modified l-Bernstein Polynomial in Terms of Basis Function.” AIMS Mathematics 9(2): 4409–4426.
• [7] Ayman-Mursaleen, M., M. Nasiruzzaman, S. K. Sharma, and Q. B. Cai. 2024. “Invariant Means and Lacunary Sequence Spaces of Order (a;b).” Demonstratio Mathematica 57: 20240003.
• [8] O¨ zger, F. 2019. “Weighed Statistical Approximation Properties of Univariate and Bivariate l-Kantorovich Operators.” Filomat 33(11): 3473–3486.
• [9] O¨ zger, F., and K. J. Ansari. 2022. “Statistical Convergence of Bivariate Generalized Bernstein Operators via Four-Dimensional Infinite Matrices.” Filomat 36(2).
• [10] Cai, Q. B., B. Y. Lian, and G. Zhou. 2018. “Approximation Properties of l-Bernstein Operators.” Journal of Inequalities and Applications 61.
• [11] Cai, Q. B., G. Zhou, and J. Li. 2019. “Statistical Approximation Properties of l-Bernstein Operators Based on q-Integers.” Open Mathematics 17: 487–498.
• [12] Acu, A. M., and I. Rasa. 2020. “Estimates for the Differences of Positive Linear Operators and Their Derivatives.” Numerical Algorithms 85: 191–208.
• [13] Mursaleen, M., and M. Nasiruzzaman. 2017. “Some Approximation Properties of Bivariate Bleimann–Butzer–Hahn Operators Based on (p,q)-Integers.” Bollettino dell’Unione Matematica Italiana 10: 271–289.
• [14] Aslan, R. 2022. “On a Stancu Form Sz´asz–Mirakjan–Kantorovich Operator Based on Shape Parameter l.” Advances in Studies of Euro–Tbilisi Mathematical Journal 15(1): 151–166.
• [15] Aslan, R., and M. Mursaleen. 2022. “Approximation by Bivariate Chlodowsky-Type Sz´asz–Durrmeyer Operators and Associated GBS Operators on Weighted Spaces.” Journal of Inequalities and Applications 2022(1): 26.
• [16] Aslan, R. 2022. “Approximation by Sz´asz–Mirakjan–Durrmeyer Operators Based on Shape Parameter l.” Communications Faculty of Sciences University of Ankara Series A1: Mathematics and Statistics 71(2): 407–421.
• [17] Aslan, Res¸at. 2021. “Some Approximation Results on l-Sz´asz–Mirakjan–Kantorovich Operators.” Fundamental Journal of Mathematics and Applications 4(3): 150–158.
• [18] Raiz, M., A. Kumar, V. N. Mishra, and N. Rao. 2022. “Dunkl Analogue of Sz´asz–Schurer Beta Operators and Their Approximation Behavior.” Mathematical Foundations of Computing 5(4): 315–330.
• [19] O¨ zger, F., R. Aslan, and M. Ersoy. 2025. “Some Approximation Results on a Class of Sza´sz–Mirakjan–Kantorovich Operators Including Non-negative Parameter a.” Numerical Functional Analysis and Optimization 46(6): 461–484.
• [20] Rao, N., M. Heshamuddin, and M. Shadab. 2019. “Approximation Properties of Bivariate Sz´asz Operators.” Filomat 33(11): 3473–3486.
• [21] Mohiuddine, S. A., T. Acar, and A. Alotaibi. 2017. “Construction of a New Family of Bernstein–Kantorovich Operators.” Mathematical Methods in the Applied Sciences 40: 7749–7759.
• [22] Mohiuddine, S. A. 2020. “Approximation by Bivariate Generalized Bernstein–Schurer Operators and Associated GBS Operators.” Advances in Differential Equations 2020(1): 1–7.
• [23] Aslan, Res¸at. 2026. “Some Approximation Properties of a-Stancu-Chlodowsky Operators”. Fundamental Journal of Mathematics and Applications 9 (1): 50-62. https://doi.org/10.33401/fujma.1857068.
• [24] Kaur, Jaspreet, Meenu Goyal, and Khursheed Ansari. 2025. “Approximation and Estimation Errors of New Kind of Laugerre and Rathore Operators”. Fundamental Journal of Mathematics and Applications 8 (4): 212-24. https://doi.org/10.33401/fujma.1814144.
• [25] Nasiruzzaman, M., N. Rao, M. Kumar, and R. Kumar. 2021. “Approximation on Bivariate Parametric Extension of Baskakov–Durrmeyer Operator.” Filomat 35: 2783–2800.
• [26] Baytunc¸, E., H. Aktuglu, and N. Mahmudov. 2023. “A New Generalization of Sz´asz–Mirakjan–Kantorovich Operators for Better Error Estimation.” Fundamental Journal of Mathematics and Applications 6(4): 194–210.
• [27] Ditzian, Z., and V. Totik. 1987. Moduli of Smoothness. Springer.
• [28] Gadjiev, A. D., and C. Orhan. 2002. “Some Approximation Theorems via Statistical Convergence.” Rocky Mountain Journal of Mathematics: 129–138.
• [29] Duman, O., M. K. Khan, and C. Orhan. 2003. “A-Statistical Convergence of Approximating Operators.” Mathematical Inequalities and Applications 6: 689–700.
• [30] Savas, E., and M. Mursaleen. 2023. “Bezier Type Kantorovich q-Baskakov Operators via Wavelets and Some Approximation Properties.” Bulletin of the Iranian Mathematical Society 49: 68.
• [31] O¨ zger, F., R. Aslan, and M. Ersoy. 2025. “Some Approximation Results on a Class of Sza´sz–Mirakjan–Kantorovich Operators Including Non-negative Parameter a.” Numerical Functional Analysis and Optimization 46(6): 481–484.
• [32] Ayman-Mursaleen, M. 2025. “Quadratic Function PreservingWavelet Type Baskakov Operators for Enhanced Function Approximation.” Computational and Applied Mathematics 44(8): 395.
• [33] M. Raiz, N. Rao, and V. N. Mishra, Sz´asz-type operators involving q-Appell polynomials, in Approximation Theory, Sequence Spaces and Applications, eds. S. A. Mohiuddine, B. Hazarika, and H. K. Nashine, Springer, pp. 187–202, 2022.
• [34] Rao, N., M. Shahzad, and N. K. Jha. 2025. “Study of Two-Dimensional a-Modified Bernstein Bivariate Operators.” Filomat 39(5): 1509–1522.
• [35] N. Rao, M. Farid, and M. Raiz, “On the Approximations and Symmetric Properties of Frobenius–Euler–S¸ims¸ek Polynomials Connecting Sz´asz Operators,” Symmetry 17(5) (2025): 648.
• [36] Rao, N., M. Farid, and N. K. Jha. 2025. “A Study of (s, m)-Stancu–Schurer as a New Generalization and Approximations.” Journal of Inequalities and Applications 2025: 104.
• [37] Ayman-Mursaleen, M., N. Rao, M. Rani, A. Kilicman, A. A. H. A. Al-Abied, and P. Malik. 2023. “A Note on Approximation of Blending Type Bernstein–Schurer–Kantorovich Operators with Shape Parameter a.” Journal of Mathematics 2023: Article ID 5245806.
• [38] Wafi, A., and N. Rao. 2019. “Sz´asz–Gamma Operators Based on Dunkl Analogue.” Iranian Journal of Science and Technology, Transactions A: Science 43: 213–223.