A New Sequence of Bernstein-Durrmeyer Operators and Their $L_p$-Approximation Behaviour

Year 2023, Volume: 11 Issue: 4, 198 - 212, 25.10.2023

Harun Çiçek , Aydın İzgi , Nadeem Rao

Abstract

The purpose of the present manuscript is to present a new sequence of Bernstein-Durrmeyer operators. First, we investigate approximation behaviour for these sequences of operators in Lebesgue Measurable space. Further, we discuss rate of convergence and order of approximation with the aid of Korovkin theorem, modulus of continuity and Peetre K-functional in $l_p$ space. Moreover, Voronovskaja type theorem is introduced to approximate a class of functions which has first and second order continuous derivatives. In the last section, numerical and graphical analysis are investigated to show better approximation behaviour for these sequences of operators.

Keywords

Rate of convergence, order of approximation;, modulus of continuity;, Bernstein-Durrmeyer operators.

References

• [1]T. Acar, A.M. Acu and N. Manav, Approximation of functions by genuine Bernstein Dur- rmeyer type operators, J. Math. Inequal. 12 (4); 975 􀀀 987; (2018):
• [2] A. M. Acu, T. Acar, V. A. Radu.: Approximation by modied U n operators. Rev. R. Acad. Ciene. Exactas Fis. Nat. Ser. A Math. racsam 113(2019) 2715-2729.
• [3] S. N. Bernstein, Demonstration du theoreme de Weierstrass fondee sur le calcul des proba- bilites, Commun. Kharkov Math. Soc. 13, 1-2, 1912 /1913.
• [4] N.L. Braha, Some properties of new modied Szasz-Mirakyan operators in polynomial weight spaces via power summability method, Bull. Math. Anal. Appl. 10:3 (2018) 53{65.
• [5] N.L. Braha, Some properties of Baskakov-Schurer-Szasz operators via power summability methods. Quaest. Math. 42 (2019), no. 10, 1411-1426.
• [6] Q. B. Cai, B. Y. Lian, G. Zhou.: Approximation Properties of 􀀀 Bernstein operators, J. Inequal. Appl. 2018 (2018) Article 61.
• [7] N. C etin, Approximation and geometric properties of complex  -Bernstein operator, Results Math. 74 , Article number: 40; (2019):
• [8] A Izg,(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.
• [9] A. Kajla and D. Miclaus, Blending Type Approximation by GBS Operators of Generalized Bernstein-Durrmeyer Type, Results Math. 73 Article number: 1, (2018) .
• [10] K. Khan and D. K. Lobiyal, Bezier curves based on Lupas (p; q)-analogue of Bernstein functions in CAGD, Journal of Computational and Applied Mathematics, 317, 458{477, (2017).
• [11] K. Khan, D. K. Lobiyal and A. Kilicman, Bezier curves and surfaces based on modied Bern- stein polynomials, Ajerbaijan Journal of Mathematics, 9 (1), 3-21, 2019.
• [12] K. Khan, D.K. Lobiyal and A. Kilicman, A de Casteljau Algorithm for Bernstein type Poly- nomials based on (p; q)-integers, Applcatons And Appled Mathematcs, 13 (2), 997{1017, 2018.
• [13] B. Lenze, On Lipschitz-type maximal functions and their smoothness spaces,Indagationes Mathematicae 91:1 (1988) 53{63.
• [14] G. G. Lorentz,Bernstein polynomials, American Mathematical Soc. (2013).
• [15] S.A. Mohiuddine, T. Acar and A. Alotaibi, Construction of a new family of Bernstein- Kantorovich operators, Math. Methods Appl. Sci. 40; 7749 􀀀 7759; (2017).
• [16] F. Schurer, Linear positive operators in approximation theory, Math. Inst. Techn. Univ. Delft Report, (1962):