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http://dx.doi.org/10.5666/KMJ.2022.62.3.467

Some Approximation Results by Bivariate Bernstein-Kantorovich Type Operators on a Triangular Domain  

Aslan, Resat (Department of Mathematics, Faculty of Sciences and Arts, Harran University)
Izgi, Aydin (Department of Mathematics, Faculty of Sciences and Arts, Harran University)
Publication Information
Kyungpook Mathematical Journal / v.62, no.3, 2022 , pp. 467-484 More about this Journal
Abstract
In this work, we define bivariate Bernstein-Kantorovich type operators on a triangular domain and obtain some approximation results for these operators. We start off by computing some moment estimates and prove a Korovkin type convergence theorem. Then, we estimate the rate of convergence using the partial and complete modulus of continuity, and derive a Voronovskaya-type asymptotic theorem. Further, we calculate the order of approximation with regard to the Peetre's K-functional and a Lipschitz type class. In addition, we construct the associated GBS type operators and compute the rate of approximation using the mixed modulus of continuity and class of the Lipschitz of Bögel continuous functions for these operators. Finally, we use the two operators to approximate example functions in order to compare their convergence.
Keywords
Bernstein-Kantorovich operators; Modulus of continuity; Voronovskaya-type asymptotic theorem; Peetre's K-functional; GBS type operators;
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1 O. T. Pop and M. D. Farcas, Approximation of b-continuous and b-differentiable functions by GBS operators of Bernstein bivariate polynomials, J. Inequal. Pure Appl. Math., 7(9)(2006).
2 O. T. Pop and M. D. Farcas, About the bivariate operators of Kantorovich type, Acta Math. Univ. Comenian., 78(1)(2009), 43-52.
3 C. Badea, K-functionals and moduli of smoothness of functions defined on compactmetric spaces, Comput. Math. with Appl., 30(1995), 23-31.   DOI
4 C. Badea, I. Badea and H. H. Gonska, Notes on degree of approximation on bcontinuous and b-differentiable functions, Approx. Theory Appl., 4(1988), 95-108.
5 G. Bascanbaz-Tunca, A. Erencin and H. Ince-Ilarslan, Bivariate Cheney-Sharma operators on simplex, Hacet. J. Math. Stat., 47(4)(2018), 793.804.
6 S. N. Bernstein, Demonstration du theor'eme de Weierstrass fondee sur le calcul des probabilites, Comp. Comm. Soc. Mat. Charkow Ser., 13(1)(1912), 1-2.
7 D. D. Stancu, A method for obtaining polynomials of Bernstein type of two variables, Am. Math. Mon., 70(3)(1963), 260-264.   DOI
8 D. X. Zhou, On smoothness characterized by Bernstein type operators, J. Approx. Theory, 81(3)(1995), 303-315.   DOI
9 L. V. Kantorovich, Sur certain developpements suivant les polynˆomes de la forme des. Bernstein, I, II, CR Acad. URSS, 563(1930), 568.
10 P. N. Agrawal, A. M. Acu, R. Chaugan and T. Garg, Approximation of Bogel Continuous functions and deferred weighted A-statistical convergence by BernsteinKantorovich type operators on a triangle, J. Math. Inequal., 15(4)(2021), 1695-1711.
11 M. Goyal, A. Kajla, P. N. Agrawal and S. Araci, Approximation by bivariate Bernstein-Durrmeyer operators on a triangle, Appl. Math. Inf. Sci., 11(2017), 693-702.   DOI
12 K. Bogel, Uber die mehrdimensionale differentiation , Jahresber. Deutsch. Math. Verein., 65(1962), 45-71.
13 E. Dobrescu and I. Matei, The approximation by Bernstein type polynomials of bidimensional continuous functions, An. Univ. Timisoara Ser. Sti. Mat.-Fiz., 4(1966), 85-90.
14 B. R. Draganov, Strong estimates of the weighted simultaneous approximation by the Bernstein and Kantorovich operators and their iterated boolean sums, J. Approx. Theory, 200(2015), 92-135.   DOI
15 A. Kajla, Generalized Bernstein-Kantorovich-type operators on a triangle, Math. Methods Appl. Sci., 42(12)(2019), 4365-4377.   DOI
16 D. Barbosu and O. T. Pop, A note on the gbs Bernstein's approximation formula, An. Univ. Craiova Ser. Mat. Inform., 35(2008), 1-6.
17 K. Bogel, Uber mehrdimensionale differentiation von funktionen mehrerer reeller veranderlichen, J. fur die Reine und Angew. Math., 170(1934), 197-217.   DOI
18 K. J. Ansari, F. Ozger and Z. Odemis Ozger, Numerical and theoretical approximation results for Schurer-Stancu operators with shape parameter λ, Comp. Appl. Math., 41, 181(2022).   DOI
19 K. Bogel, Uber mehrdimensionale differentiation, integration und beschrankte variation, J. fur die Reine und Angew. Math., 173(1935), 5-30.   DOI
20 A. Kajla and M. Goyal, Modified Bernstein-Kantorovich operators for functions of one and two variables, Rend. Circ. Mat. Palermo(2), 67(2)(2018), 379-395.   DOI
21 S. Deshwal, N. Ispir and P. N. Agrawal, Bivariate operators of Bernstein-Kantorovich type on a triangle, Appl. Math. Inf. Sci., 11(2)(2017), 423-432.   DOI
22 S. G. Gal, Shape-preserving approximation by real and complex polynomials, Springer Science and Business Media(2010).
23 M. Sidharth, N. Ispir and P. N. Agrawal, GBS operators of Bernstein-Schurer-Kantorovich type based on q-integers, Appl. Math. Comput., 269(2015), 558-568.   DOI
24 H. Srivastava, K. J. Ansari, F. Ozger and Z. Odemis Ozger, A link between approximation theory and summability methods via four-dimensional infinite matrices. Mathematics 9:1895 (2021).   DOI
25 V. I. Volkov, On the convergence of sequences of linear positive operators in the space of continuous functions of two variables, Dokl. Akad. Nauk SSSR, 151(1)(1957), 17-19.
26 K. Weierstrass, Uber die analytische darstellbarkeit sogenannter willkurlicher functionen einer reellen veranderlichen, Sitzungsberichte der Koniglich Preussischen Akademie der Wissenschaften zu Berlin, 2(1885), 633-639.
27 E. H. Kingsley, Bernstein polynomials for functions of two variables of class C(k), Proc. Amer. Math. Soc., 2(1)(1951), 64-71.   DOI
28 M. Heshamuddin, N. Rao, B. P. Lamichhane, A. Kilicman and M. Ayman-Mursaleen, On One- and Two-Dimensional α-Stancu-Schurer-Kantorovich Operators and Their Approximation Properties. Mathematics, 10(18), 3227 (2022).   DOI
29 R. Ruchi, B. Baxhaku and P. N. Agrawal, GBS operators of bivariate BernsteinDurrmeyer type on a triangle, Math. Methods Appl. Sci., 41(7)(2018), 2673-2683.   DOI