On the Korovkin-Type and Statistical Approximation of A New Linear Positive Operators

Aydın İzgi; Döne Karahan Dinsever

doi:10.31801/cfsuasmas.1468206

On the Korovkin-Type and Statistical Approximation of A New Linear Positive Operators

Abstract

In this paper, a new generalization of Bernstein operators is defined. The Korovkin type approximation and the statistical approximation of this new operator are studied. Voronovskaja type and quantitative Voronovskaja type approximation theorems are proved. Graphs of the approximation are obtained using the Maple program.

Keywords

References

Acar, E., Güler, Ö., and Serenbay, S. K. (2023). Approximation by nonlinear Bernstein-Chlodowsky operators of Kantorovich type. Filomat, 37(14):4621–4627.
Altomare, F. and Campiti, M. (1994). Korovkin Type Approximation Theory and Its Applications. de Gruyter, Berlin and New York.
Ari, D. A. and Serenbay, S. K. (2019). Approximation by a generalization of the Jakimovski-Leviatan operators. Filomat, 33(8):2345–2353.
Bernstein, S. (1912). Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités. Communications of the Mathematical Society of Kharkov, 13(2):1–2.
Bohman, H. (1952). On approximation of continuous and of analytic functions. Arkiv för Matematik, 2:43–56.
Braha, N. L., Loku, V., Mansour, T., and Mursaleen, M. (2022). A new weighted statistical convergence and some associated approximation theorems. Mathematical Methods in the Applied Sciences, 45:5682– 5698.
Çilo, A. (2012). $[−1, 1]$ aralığında Bernstein polinomlarının yaklaşım özellikleri ve yaklaşım hızı. Master’s thesis, Harran University.
Ditzian, Z. and Totik, V. (1987). Moduli of Smoothness, volume 9 of Springer Series in Computational Mathematics. Springer-Verlag, New York.

Erdős, P. and Tenenbaum, G. (1989). Sur les densités de certaines suites d’entiers. Proceedings of the London Mathematical Society, 59(3):417–438.
Fast, H. (1951). Sur la convergence statistique. Colloquium Mathematicum, 2:241–244.
Finta, Z. (2011). Remark on Voronovskaja theorem for $q$-Bernstein operators. Studia Universitatis Babeş-Bolyai Mathematica, 56:335–339.
Gadjiev, A. D. and Orhan, C. (2002). Some approximation theorems via statistical convergence. Rocky Mountain Journal of Mathematics, 32:129–138.
Güller, Ö., Acar, E., and Serenbay, S. K. (2022). Nonlinear bivariate Bernstein-Chlodowsky operators of maximum product type. Journal of Mathematics, page 11.
İzgi, A. (2012). Approximation by a class of new type Bernstein polynomials of one and two variables. Global Journal of Pure and Applied Mathematics, 8(5):55–71.
Korovkin, P. P. (1953). On the convergence of linear positive operators in the space of continuous functions. Doklady Akademii Nauk SSSR, 90:961–964. (Russian).
Popoviciu, T. (1950a). Asupra demonstratiei teoremei lui Weierstrass cu ajutorul polinoamelor de interpolare. Lucrările Sesiunii Generale Stiintifice, pages 2–12.
Popoviciu, T. (1950b). Asupra formei restului în unele formule de approximatie ale analizei. Lucrările sesiunii generale ştiintifice a Acad. R.P.R., pages 183–196.
Raiz, M., Kumar, A., Mishra, V. N., and Rao, N. (2022). Dunkl analogue of Szász-Schurer-Beta operators and their approximation behaviour. Mathematical Foundations of Computing, 5(4):315–330.
Rao, N., Raiz, M., and Ayman-Mursaleen, M. (2023). Approximation properties of extended beta-type Szász–Mirakjan operators. Iranian Journal of Science, 47:1771–1781.
Schoenberg, I. J. (1959). The integrability of certain functions and related summability methods. American Mathematical Monthly, 66:361–375.
Steinhaus, H. (1951). Sur la convergence ordinaire et la convergence asymptotique. Colloquium Mathematicum, 2:73–74.
Vernescu, A. (2009). Revisiting the Popoviciu-Bohman-Korovkin theorem. Journal of Science and Arts, 2(11):237–240.

Details

Primary Language

English

Subjects

Approximation Theory and Asymptotic Methods

Journal Section

Research Article

Authors

Aydın İzgi
0000-0003-3715-8621
Türkiye

Döne Karahan Dinsever ^*
0000-0001-6644-5596
Türkiye

Publication Date

March 29, 2026

Submission Date

April 14, 2024

Acceptance Date

July 16, 2025

Published in Issue

Year 1970 Volume: 75 Number: 1

DOI

https://doi.org/10.31801/cfsuasmas.1468206

IZ

https://izlik.org/JA96EN48PD

Cite

RIS / Bibtex

APA

İzgi, A., & Karahan Dinsever, D. (2026). On the Korovkin-Type and Statistical Approximation of A New Linear Positive Operators. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 75(1), 14-25. https://doi.org/10.31801/cfsuasmas.1468206

AMA

1.İzgi A, Karahan Dinsever D. On the Korovkin-Type and Statistical Approximation of A New Linear Positive Operators. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2026;75(1):14-25. doi:10.31801/cfsuasmas.1468206

Chicago

İzgi, Aydın, and Döne Karahan Dinsever. 2026. “On the Korovkin-Type and Statistical Approximation of A New Linear Positive Operators”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 75 (1): 14-25. https://doi.org/10.31801/cfsuasmas.1468206.

EndNote

İzgi A, Karahan Dinsever D (March 1, 2026) On the Korovkin-Type and Statistical Approximation of A New Linear Positive Operators. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 75 1 14–25.

IEEE

[1]A. İzgi and D. Karahan Dinsever, “On the Korovkin-Type and Statistical Approximation of A New Linear Positive Operators”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 75, no. 1, pp. 14–25, Mar. 2026, doi: 10.31801/cfsuasmas.1468206.

ISNAD

İzgi, Aydın - Karahan Dinsever, Döne. “On the Korovkin-Type and Statistical Approximation of A New Linear Positive Operators”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 75/1 (March 1, 2026): 14-25. https://doi.org/10.31801/cfsuasmas.1468206.

JAMA

1.İzgi A, Karahan Dinsever D. On the Korovkin-Type and Statistical Approximation of A New Linear Positive Operators. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2026;75:14–25.

MLA

İzgi, Aydın, and Döne Karahan Dinsever. “On the Korovkin-Type and Statistical Approximation of A New Linear Positive Operators”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 75, no. 1, Mar. 2026, pp. 14-25, doi:10.31801/cfsuasmas.1468206.

Vancouver

1.Aydın İzgi, Döne Karahan Dinsever. On the Korovkin-Type and Statistical Approximation of A New Linear Positive Operators. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2026 Mar. 1;75(1):14-25. doi:10.31801/cfsuasmas.1468206