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Mathematics
Volume 13
Issue 22
10.3390/math13223586
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Article

Convergence by Class of Kantorovich-Type q-Szász Operators and Comprehensive Results

by
Md. Nasiruzzaman
1,
Mohammad Farid
2,* and
Nadeem Rao
3
1
Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 4279, Tabuk 71491, Saudi Arabia
2
Department of Mathematics, College of Science, Qassim University, Saudi Arabia
3
Department of Mathematics, University Center for Research and Development, Chandigarh University, Mohali 140413, Punjab, India
*
Author to whom correspondence should be addressed.
Mathematics 2025, 13(22), 3586; https://doi.org/10.3390/math13223586 (registering DOI)
Submission received: 21 September 2025 / Revised: 1 November 2025 / Accepted: 3 November 2025 / Published: 8 November 2025
(This article belongs to the Special Issue Advances in Functional Analysis and Approximation Theory)
Versions Notes

Abstract

In this paper, we primarily use Stancu variants of Kantorovich-type operators to investigate the convergence and other associated properties of new Szász–Mirakjan operators. We compute the moments and central moments of the new Szász–Mirakjan operators by q-integers and propose their modified Kantorovich form. More specifically, we examine the convergence characteristics in the space of continuous functions. With the use of the modulus of continuity and the integral modulus of continuity, we determine the degree of convergence. Additionally, we obtain the Voronovskaja type theorems. To validate convergence, we conclude with a numerical example and graphical illustration of the operator sequences.
Keywords: Szász–Mirakjan operators; q-integrals; Korovkin’s theorem; modulus of continuity; Peetre’s K-functional; approximation algorithms; Voronovskaja-type theorem; mathematical operators Szász–Mirakjan operators; q-integrals; Korovkin’s theorem; modulus of continuity; Peetre’s K-functional; approximation algorithms; Voronovskaja-type theorem; mathematical operators

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MDPI and ACS Style

Nasiruzzaman, M.; Farid, M.; Rao, N. Convergence by Class of Kantorovich-Type q-Szász Operators and Comprehensive Results. Mathematics 2025, 13, 3586. https://doi.org/10.3390/math13223586

AMA Style

Nasiruzzaman M, Farid M, Rao N. Convergence by Class of Kantorovich-Type q-Szász Operators and Comprehensive Results. Mathematics. 2025; 13(22):3586. https://doi.org/10.3390/math13223586

Chicago/Turabian Style

Nasiruzzaman, Md., Mohammad Farid, and Nadeem Rao. 2025. "Convergence by Class of Kantorovich-Type q-Szász Operators and Comprehensive Results" Mathematics 13, no. 22: 3586. https://doi.org/10.3390/math13223586

APA Style

Nasiruzzaman, M., Farid, M., & Rao, N. (2025). Convergence by Class of Kantorovich-Type q-Szász Operators and Comprehensive Results. Mathematics, 13(22), 3586. https://doi.org/10.3390/math13223586

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