A new generalization of Szasz-Kantorovich operators on weighted space

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.

Keywords

• Linear positive operators
• Szasz-­Kantorovich operators
• ­Weighted space
• Module of ­Continuity
• Lipschitz class

Kaynakça

1. [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. [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. [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. [4] Bohman,H.(1952-54). On approximation of continuous and analytic functions. Ark. Math.,2, 43-46.
5. [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. [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. [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. [8] Kantorovich ,L. V.(1930). Sur cestain developpemenets suivant les polynomes de la forme de S.Berntein. C.R. Acad.,595-600.

9. [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. [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. [11] Korovkin, P.P.(1930). On Convergence of Linear Positive Operators in the Space of Continuous Functions. Dokl. Akad. Nauk, 90, 961-964.
12. [12] Lenze,B.(1988). On Lipschitz-type maximal functions and their smoothness spaces.Indagationes Mathematicae, 91:1 53–63.
13. [13] Lorentz,G. G.(2013).Bernstein polynomials, American Mathematical Soc.
14. [14] Schurer,F. (1962). Linear positive operators in approximation theory. Math. Inst. Techn. Univ. Delft Report.
15. [15] Szasz,O.(1950). Genelalization of S. Bernstein’s polynomials to the infinite interval. J. Research Nat. Bur. Standars, 45, 239-245.
16. [16] Totik,V.(1983). Approximation by Szasz-Mirakjan-Kantorovich operators in Lp space. Analysis Mathematica, 9(2), 147-167.
17. [17] Walzack,Z.(2000). On certain Moddified Szasz-Mirakjan operators for functions of two variables. Demonstratio Math ., 31(1), 91-100.
18. [18] Weirestrass,K.(1885). Uber die Analytische Darstellbarkeit Sogenannter Willk ¨ urlicher Funktionen Einer Reelen Ver ¨ anderlichen. ¨ Sitzungberichte der Akademie zu Berlin, 789-805.