Abstract
In this work, we construct a new kind of Bernstein–Schurer operators which includes non-negative real parameter \(\alpha \). We study some shape preserving properties, namely, monotonicity and convexity of the new operators. We obtain global approximation formula in terms of Ditzian–Totik uniform modulus of smoothness of first and second order and calculate the local direct estimate of the rate of convergence with the help of Lipschitz-type function for our operators. The Voronovskaja-type approximation theorems of the new operators are presented. Finally, in the last section, we provide some graphs, MATLAB code and numerical examples in order to illustrate the significance of our newly constructed operators.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Acu, A., Acar, T., Radu, V.A.: Approximation by modified \(U_{n}^{\rho }\) operators. Rev. R. Acad. Cienc. Exactas Fıs. Nat. Ser. A Math. RACSAM 113, 2715–2729 (2019)
Agrawal, P.N., Acu, A.M., Sidharth, M.: Approximation degree of a Kantorovich variant of Stancu operators based on Pólya–Eggenberger distribution. Rev. R. Acad. Cienc. Exactas Fıs. Nat. Ser. A Math. RACSAM 113, 137–156 (2019)
Barbosu, D.: The Voronovskaja theorem for Bernstein–Schurer operators. Bul. Ştiinţ. Univ. Baia Mare Ser. B Matematică Informatică 18(2), 137–140 (2002)
Bernstein, S.N.: Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités. Commun. Kharkov Math. Soc. 13, 1–2 (1912/1913)
Cai, Q.-B.: The Bézier variant of Kantorovich type \(\lambda \)-Bernstein operators. J. Inequal. Appl. 2018, Article 90 (2018)
Cai, Q.B., Lian, B.Y., Zhou, G.: Approximation properties of \(\lambda \)-Bernstein operators. J. Inequal. Appl. 2018, Article 61 (2018)
Chen, X., Tan, J., Liu, Z., Xie, J.: Approximation of functions by a new family of generalized Bernstein operators. J. Math. Anal. Appl. 450, 244–261 (2017)
DeVore, R.A., Lorentz, G.G.: Constructive Approximation. Springer, Berlin (1993)
Ditzian, Z., Totik, V.: Moduli of Smoothness. Springer, New York (1987)
Kajla, A., Acar, T.: Blending type approximation by generalized Bernstein–Durrmeyer type operators. Miskolc Math. Notes 19, 319–336 (2018)
Kajla, A., Acar, T.: Bézier-Bernstein–Durrmeyer type operators. Rev. R. Acad. Cienc. Exactas Fıs. Nat. Ser. A Math. RACSAM 114, 31 (2020)
Kajla, A., Miclăuş, D.: Blending type approximation by GBS operators of generalized Bernstein-Durrmeyer type. Results Math. 73, 1 (2018)
Mohiuddine, S.A., Acar, T.: Advances in Summability and Approximation Theory, Springer (2018)
Mohiuddine, S.A., Acar, T., Alghamdi, M.A.: Genuine modified Bernstein–Durrmeyer operators. J. Inequal. Appl. 2018, Article 104 (2018)
Mohiuddine, S.A., Acar, T., Alotaibi, A.: Construction of a new family of Bernstein–Kantorovich operators. Math. Methods Appl. Sci. 40, 7749–7759 (2017)
Mohiuddine, S.A., Alamri, B.A.S.: Generalization of equi-statistical convergence via weighted lacunary sequence with associated Korovkin and Voronovskaya type approximation theorems. Rev. R. Acad. Cienc. Exactas Fıs. Nat. Ser. A Math. RACSAM 113(3), 1955–1973 (2019)
Mohiuddine, S.A., Asiri, A., Hazarika, B.: Weighted statistical convergence through difference operator of sequences of fuzzy numbers with application to fuzzy approximation theorems. Int. J. Gen. Syst. 48(5), 492–506 (2019)
Mohiuddine, S.A., Hazarika, B., Alghamdi, M.A.: Ideal relatively uniform convergence with Korovkin and Voronovskaya types approximation theorems. Filomat 33(14), 4549–4560 (2019)
Mohiuddine, S.A., Özger, F.: Approximation of functions by Stancu variant of Bernstein–Kantorovich operators based on shape parameter \(\alpha \). Rev. R. Acad. Cienc. Exactas Fıs. Nat. Ser. A Math. RACSAM 114, 70 (2020)
Mursaleen, M., Ansari, K.J., Khan, A.: Approximation by a Kantorovich type \(q\)-Bernstein–Stancu operators. Complex Anal. Oper. Theory 11(1), 85–107 (2017)
Mursaleen, M., Khan, F., Khan, A.: Approximation properties for King’s type modified \(q\)-Bernstein–Kantorovich operators. Math. Methods Appl. Sci. 38, 5242–5252 (2015)
Nasiruzzaman, M., Rao, N., Wazir, S., Kumar, R.: Approximation on parametric extension of Baskakov–Durrmeyer operators on weighted spaces. J. Inequal. Appl. 2019, Article 103 (2019)
Ozarslan, M.A., Aktuğlu, H.: Local approximation for certain King type operators. Filomat 27, 173–181 (2013)
Özger, F.: Weighted statistical approximation properties of univariate and bivariate \(\lambda \)-Kantorovich operators. Filomat 33(11), 1–15 (2019)
Özger, F.: On new Bézier bases with Schurer polynomials and corresponding results in approximation theory. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 69(1), 1–18 (2020)
Schurer, F.: Linear positive operators in approximation theory. Math. Inst. Techn. Univ, Delft Report (1962)
Srivastava, H.M.: Operators of basic (or \(q\)-) calculus and fractional \(q\)-calculus and their applications in geometric function theory of complex analysis. Iran. J. Sci. Technol. Trans. A Sci. 44, 327–344 (2020)
Srivastava, H.M., Özger, F., Mohiuddine, S.A.: Construction of Stancu-type Bernstein operators based on Bézier bases with shape parameter \(\lambda \). Symmetry 11(3), 316 (2019)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Özger, F., Srivastava, H.M. & Mohiuddine, S.A. Approximation of functions by a new class of generalized Bernstein–Schurer operators. RACSAM 114, 173 (2020). https://doi.org/10.1007/s13398-020-00903-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13398-020-00903-6
Keywords
- Bernstein operators
- \(\alpha \)-Bernstein–Schurer
- Shape preserving property
- Uniform convergence
- Global approximation
- Voronovskaja-type approximation theorem