Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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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)
  • Received : 2021.02.03
  • Accepted : 2022.01.24
  • Published : 2022.09.30
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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.

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References

  1. 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.
  2. 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). https://doi.org/10.1007/s40314-022-01877-4
  3. C. Badea, K-functionals and moduli of smoothness of functions defined on compactmetric spaces, Comput. Math. with Appl., 30(1995), 23-31. https://doi.org/10.1016/0898-1221(95)00083-6
  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. 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.
  6. G. Bascanbaz-Tunca, A. Erencin and H. Ince-Ilarslan, Bivariate Cheney-Sharma operators on simplex, Hacet. J. Math. Stat., 47(4)(2018), 793.804.
  7. 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.
  8. K. Bogel, Uber mehrdimensionale differentiation von funktionen mehrerer reeller veranderlichen, J. fur die Reine und Angew. Math., 170(1934), 197-217. https://doi.org/10.1515/crll.1934.170.197
  9. K. Bogel, Uber mehrdimensionale differentiation, integration und beschrankte variation, J. fur die Reine und Angew. Math., 173(1935), 5-30. https://doi.org/10.1515/crll.1935.173.5
  10. K. Bogel, Uber die mehrdimensionale differentiation , Jahresber. Deutsch. Math. Verein., 65(1962), 45-71.
  11. 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. https://doi.org/10.18576/amis/110210
  12. 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.
  13. 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. https://doi.org/10.1016/j.jat.2015.07.006
  14. S. G. Gal, Shape-preserving approximation by real and complex polynomials, Springer Science and Business Media(2010).
  15. 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. https://doi.org/10.18576/amis/110308
  16. A. Kajla, Generalized Bernstein-Kantorovich-type operators on a triangle, Math. Methods Appl. Sci., 42(12)(2019), 4365-4377. https://doi.org/10.1002/mma.5656
  17. 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. https://doi.org/10.1007/s12215-017-0320-z
  18. L. V. Kantorovich, Sur certain developpements suivant les polynˆomes de la forme des. Bernstein, I, II, CR Acad. URSS, 563(1930), 568.
  19. E. H. Kingsley, Bernstein polynomials for functions of two variables of class C(k), Proc. Amer. Math. Soc., 2(1)(1951), 64-71. https://doi.org/10.1090/S0002-9939-1951-0042548-7
  20. 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). https://doi.org/10.3390/math10183227
  21. 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).
  22. O. T. Pop and M. D. Farcas, About the bivariate operators of Kantorovich type, Acta Math. Univ. Comenian., 78(1)(2009), 43-52.
  23. 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. https://doi.org/10.1002/mma.4771
  24. 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. https://doi.org/10.1016/j.amc.2015.07.052
  25. 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). https://doi.org/10.3390/math9161895
  26. D. D. Stancu, A method for obtaining polynomials of Bernstein type of two variables, Am. Math. Mon., 70(3)(1963), 260-264. https://doi.org/10.2307/2313121
  27. 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.
  28. 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.
  29. D. X. Zhou, On smoothness characterized by Bernstein type operators, J. Approx. Theory, 81(3)(1995), 303-315. https://doi.org/10.1006/jath.1995.1052