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A new generalization of Szasz-Kantorovich operators on weighted space

Year 2022, Volume: 7 Issue: 2, 85 - 106, 30.09.2022

Abstract

The purpose of this article is to define a new generalization of Szász-Kantorovich operators. First, by using the Korovkin theorem on the new operator we define, its convergence properties and rates are examined. Then, the Voronovskaja-type theorem for the new operator is proven. Additionally, with the help of the modulus of continuity in the weighted space, rate of convergence the new operator is examined, and a theorem is proven for the operator we define by using functions that satisfy the Lipschitz condition. Finally, the convergence is demonstrated more clearly by numerical examples and plots.

References

  • [1] Acar, T., Acu, A.M. and Manav,N.(2018). Approximation of functions by genuine Bernstein Durrmeyer type operators. J. Math. Inequal, 12 (4), 975 − 987.
  • [2] Acu,A. M., Acar,T., Radu.V. A.(2019).Approximation by modified Un operators. Rev. R. Acad. Ciene. Exactas Fis. Nat. Ser. A Math.i 113-2715-2729.
  • [3] Bernstein,S. N.(1913). Demonstration du th ´ eor ´ eme de Weierstrass fond ´ ee sur le calcul des probabilit ´ es. Commun. Kharkov Math. ´ Soc., 13, 1-2.
  • [4] Bohman,H.(1952-54). On approximation of continuous and analytic functions. Ark. Math.,2, 43-46.
  • [5] Chlodovsky,I.(1937). Sur le developpement des fonctions definies dans un intervalle infini en series de polynomes de M.S. Bernstein. Compos Math,4, 380-393.
  • [6] Gupta,V. , Vasishtha,V. (2002). Rate of convergene of the Szasz-Kantorovich-Bezier operators for bounded variation function. Publ. Inst. Math.,72,137-143.
  • [7] Izgı,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.
  • [8] Kantorovich ,L. V.(1930). Sur cestain developpemenets suivant les polynomes de la forme de S.Berntein. C.R. Acad.,595-600.
  • [9] Kajla,A., Miclaus¸,D. (2018). Blending Type Approximation by GBS Operators of ˇ Generalized Bernstein-Durrmeyer Type. Results Math., 73 Article number: 1 .
  • [10] Khan,K., Lobiyal,D. K., Kilicman,A. (2019). Bezier curves and surfaces based on modified Bernstein polynomial ` s. Ajerbaijan Journal of Mathematics, 9 (1), 3-21.
  • [11] Korovkin, P.P.(1930). On Convergence of Linear Positive Operators in the Space of Continuous Functions. Dokl. Akad. Nauk, 90, 961-964.
  • [12] Lenze,B.(1988). On Lipschitz-type maximal functions and their smoothness spaces.Indagationes Mathematicae, 91:1 53–63.
  • [13] Lorentz,G. G.(2013).Bernstein polynomials, American Mathematical Soc.
  • [14] Schurer,F. (1962). Linear positive operators in approximation theory. Math. Inst. Techn. Univ. Delft Report.
  • [15] Szasz,O.(1950). Genelalization of S. Bernstein’s polynomials to the infinite interval. J. Research Nat. Bur. Standars, 45, 239-245.
  • [16] Totik,V.(1983). Approximation by Szasz-Mirakjan-Kantorovich operators in Lp space. Analysis Mathematica, 9(2), 147-167.
  • [17] Walzack,Z.(2000). On certain Moddified Szasz-Mirakjan operators for functions of two variables. Demonstratio Math ., 31(1), 91-100.
  • [18] Weirestrass,K.(1885). Uber die Analytische Darstellbarkeit Sogenannter Willk ¨ urlicher Funktionen Einer Reelen Ver ¨ anderlichen. ¨ Sitzungberichte der Akademie zu Berlin, 789-805.
Year 2022, Volume: 7 Issue: 2, 85 - 106, 30.09.2022

Abstract

References

  • [1] Acar, T., Acu, A.M. and Manav,N.(2018). Approximation of functions by genuine Bernstein Durrmeyer type operators. J. Math. Inequal, 12 (4), 975 − 987.
  • [2] Acu,A. M., Acar,T., Radu.V. A.(2019).Approximation by modified Un operators. Rev. R. Acad. Ciene. Exactas Fis. Nat. Ser. A Math.i 113-2715-2729.
  • [3] Bernstein,S. N.(1913). Demonstration du th ´ eor ´ eme de Weierstrass fond ´ ee sur le calcul des probabilit ´ es. Commun. Kharkov Math. ´ Soc., 13, 1-2.
  • [4] Bohman,H.(1952-54). On approximation of continuous and analytic functions. Ark. Math.,2, 43-46.
  • [5] Chlodovsky,I.(1937). Sur le developpement des fonctions definies dans un intervalle infini en series de polynomes de M.S. Bernstein. Compos Math,4, 380-393.
  • [6] Gupta,V. , Vasishtha,V. (2002). Rate of convergene of the Szasz-Kantorovich-Bezier operators for bounded variation function. Publ. Inst. Math.,72,137-143.
  • [7] Izgı,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.
  • [8] Kantorovich ,L. V.(1930). Sur cestain developpemenets suivant les polynomes de la forme de S.Berntein. C.R. Acad.,595-600.
  • [9] Kajla,A., Miclaus¸,D. (2018). Blending Type Approximation by GBS Operators of ˇ Generalized Bernstein-Durrmeyer Type. Results Math., 73 Article number: 1 .
  • [10] Khan,K., Lobiyal,D. K., Kilicman,A. (2019). Bezier curves and surfaces based on modified Bernstein polynomial ` s. Ajerbaijan Journal of Mathematics, 9 (1), 3-21.
  • [11] Korovkin, P.P.(1930). On Convergence of Linear Positive Operators in the Space of Continuous Functions. Dokl. Akad. Nauk, 90, 961-964.
  • [12] Lenze,B.(1988). On Lipschitz-type maximal functions and their smoothness spaces.Indagationes Mathematicae, 91:1 53–63.
  • [13] Lorentz,G. G.(2013).Bernstein polynomials, American Mathematical Soc.
  • [14] Schurer,F. (1962). Linear positive operators in approximation theory. Math. Inst. Techn. Univ. Delft Report.
  • [15] Szasz,O.(1950). Genelalization of S. Bernstein’s polynomials to the infinite interval. J. Research Nat. Bur. Standars, 45, 239-245.
  • [16] Totik,V.(1983). Approximation by Szasz-Mirakjan-Kantorovich operators in Lp space. Analysis Mathematica, 9(2), 147-167.
  • [17] Walzack,Z.(2000). On certain Moddified Szasz-Mirakjan operators for functions of two variables. Demonstratio Math ., 31(1), 91-100.
  • [18] Weirestrass,K.(1885). Uber die Analytische Darstellbarkeit Sogenannter Willk ¨ urlicher Funktionen Einer Reelen Ver ¨ anderlichen. ¨ Sitzungberichte der Akademie zu Berlin, 789-805.
There are 18 citations in total.

Details

Primary Language English
Journal Section Volume VII Issue II
Authors

Harun Çiçek 0000-0003-3018-3015

Shaymaa Jameel Zainalabdin 0000-0002-3861-3430

Aydın İzgi 0000-0003-3715-8621

Publication Date September 30, 2022
Published in Issue Year 2022 Volume: 7 Issue: 2

Cite

APA Çiçek, H., Jameel Zainalabdin, S., & İzgi, A. (2022). A new generalization of Szasz-Kantorovich operators on weighted space. Turkish Journal of Science, 7(2), 85-106.
AMA Çiçek H, Jameel Zainalabdin S, İzgi A. A new generalization of Szasz-Kantorovich operators on weighted space. TJOS. September 2022;7(2):85-106.
Chicago Çiçek, Harun, Shaymaa Jameel Zainalabdin, and Aydın İzgi. “A New Generalization of Szasz-Kantorovich Operators on Weighted Space”. Turkish Journal of Science 7, no. 2 (September 2022): 85-106.
EndNote Çiçek H, Jameel Zainalabdin S, İzgi A (September 1, 2022) A new generalization of Szasz-Kantorovich operators on weighted space. Turkish Journal of Science 7 2 85–106.
IEEE H. Çiçek, S. Jameel Zainalabdin, and A. İzgi, “A new generalization of Szasz-Kantorovich operators on weighted space”, TJOS, vol. 7, no. 2, pp. 85–106, 2022.
ISNAD Çiçek, Harun et al. “A New Generalization of Szasz-Kantorovich Operators on Weighted Space”. Turkish Journal of Science 7/2 (September 2022), 85-106.
JAMA Çiçek H, Jameel Zainalabdin S, İzgi A. A new generalization of Szasz-Kantorovich operators on weighted space. TJOS. 2022;7:85–106.
MLA Çiçek, Harun et al. “A New Generalization of Szasz-Kantorovich Operators on Weighted Space”. Turkish Journal of Science, vol. 7, no. 2, 2022, pp. 85-106.
Vancouver Çiçek H, Jameel Zainalabdin S, İzgi A. A new generalization of Szasz-Kantorovich operators on weighted space. TJOS. 2022;7(2):85-106.