A New Generalization of Szász-Mirakjan Kantorovich Operators for Better Error Estimation

Year 2023, , 194 - 210, 31.12.2023

Erdem Baytunç , Hüseyin Aktuğlu , Nazım Mahmudov

https://doi.org/10.33401/fujma.1355254

Cited By: 4

Abstract

In this article, we construct a new sequence of Szász-Mirakjan-Kantorovich operators denoted as $K_{n,\gamma}(f;x)$, which depending on a parameter $\gamma$. We prove direct and local approximation properties of $K_{n,\gamma}(f;x)$. We obtain that, if $\gamma>1$, then the operators $K_{n,\gamma}(f;x)$ provide better approximation results than classical case for all $x\in[0,\infty)$. Furthermore, we investigate the approximation results of $K_{n,\gamma}(f;x)$, graphically and numerically. Moreover, we introduce new operators from $K_{n,\gamma}(f;x)$ that preserve affine functions and bivariate case of $K_{n,\gamma}(f;x)$. Then, we study their approximation properties and also illustrate the convergence of these operators comparing with their classical cases.

Keywords

Affine functions, Bivariate Szasz-Mirakjan Kantorovich operators, ´ Modulus of continuity, Positive Linear Operators, Rate of convergence, Szasz- ´ Mirakjan Kantorovich operator

References

• [1] V.K. Weierstrass, Uber die analytische Darstellbarkeit sogennanter willku¨rlicher Functionen einer reellen Veranderlichen, Sitzungsberichte der Akademie zu Berlin, (1885), 633–639.
• [2] S. Bernstein, Damonstration du thaoreme de weirstrass. founde´a sur le calcul des probabilit´as., Commun. Soc. Math. Kharkow (2),(1912), 1-2.
• [3] G.G. Lorentz, Bernstein Polinomials, Chelsea, New York, (1986).
• [4] H. Aktuğlu, H. Gezer, E. Baytunç and M.S Atamert, Approximation properties of generalized blending type Lototsky-Bernstein Operators, J. Math. Inequal. 16(2) (2022), 707-728.
• [5] E. Baytunç, H. Aktuğlu and N.I. Mahmudov, Approximation properties of Riemann-Liouville type fractional Bernstein-Kantorovich operators of order $\alpha$, Math. Found. Comput., (2023).
• [6] Q.B. Cai, The Bezier variant of Kantorovich type $\lambda$-Bernstein operators, J. Inequal. Appl. 201890 (2018), 1-10.
• [7] O. Duman, M.A. Özarslan, and B.D. Vecchia, Modified Sza´sz-Mirakjan-Kantorovich operators preserving linear functions, Turk. J. Math., 33(2) (2009), 151-158.
• [8] H. Gezer, H. Aktuğlu, E. Baytunc¸ and M.S. Atamert, Generalized blending type Bernstein operators based on the shape parameter $\lambda$, J. Inequal. Appl., 2022(96) (2022), 1-19.
• [9] L.V. Kantorovich, Sur certains developements suivant les polynˆomes de la forme de S. Bernstein I, II. Dokl. Akad. Nauk SSSR 563(568) (1930), 595–600.
• [10] R. Özçelik, E.E. Kara, F. Usta, and K.J Ansari, Approximation properties of a new family of Gamma operators and their applications, Adv. Differ. Equ., 2021 508 (2021), 1-13.
• [11] G.M. Mirakjan, Approximation of continuous functions with the aid of polynomials, Dokl. Akad. Nauk SSSR, 31 (1941), 201–205.
• [12] O. Szasz, Generalization of S.Bernstein’s polynomials to the infinite interval, J. Res. Nation. Bureau Stand., Sec. B, 45 (1950), 239– 245.
• [13] P.L. Butzer, On the extension of Bernstein polynomials to the infinite interval, Proc. Am. Math. Soc., 5 (1954), 547-553.
• [14] P.N. Agrawal and H.S. Kasana, On simultaneous approximation by Sz´asz-Mirakjan operators, Bull. Inst. Math., Acad. Sin. 22 (1994), 181- 188.
• [15] V. Gupta, V. Vasishtha and M.K Gupta, Rate of convergence of the Sz´asz-Kantorovich-Bezier operators for bounded variation functions, Publ. Math. Inst. 72 (2006), 137-143.
• [16] V. Gupta and M.A. Noor, Convergence of derivatives for certain mixed Sz´asz-Beta operators, J. Math. Anal. Appl., 321(1) (2006), 1-9.
• [17] N. Mahmudov and V. Gupta, On certain $q$-analogue of Szasz Kantorovich operators, J. Appl. Math. Comput., 37 (2011), 407–419.
• [18] Q. Razi and S. Umar,$L_p$-approximation by Szasz-Mirakyan-Kantorovich operators, Indian J. Pure Appl. Math., 18(2) (1987),173-177.
• [19] M. Raiz, A. Kumar, V.N. Mishra and N. Rao, Dunkl analogue of Szasz-Schurer-Beta operators and their approximation behaviour, Math. Found. Comput., 5(4) (2022), 315-330.
• [20] M. Raiz, R.S. Rajawat and V.N. Mishra, $\alpha$-Schurer Durrmeyer operators and their approximation properties, Ann. Univ. Craiova Math. Comput. Sci. Ser., 50(1) (2023), 189-204.
• [21] A.M. Acu, I.C. Buscu and I. Raşa, Generalized Kantorovich modifications of positive linear operators, Math. Found. Comput., 6(1) (2023), 54-62.
• [22] O. Agratini, Kantorovich-type operators preserving affine functions, Hacet. J. Math. Stat., 45(6) (2016), 1657-1663.
• [23] K.J. Ansari, On Kantorovich variant of Baskakov type operators preserving some functions, Filomat, 36(3) (2022) , 1049-1060.
• [24] K.J. Ansari, M. Civelek and F. Usta, Jain’s operator: a new construction and applications in approximation theory, Math. Methods Appl. Sci., 46 (2023), 14164–14176.
• [25] K.J. Ansari and F. Usta, A Generalization of Szasz–Mirakyan operators based on $\alpha$ non-negative parameter, Symmetry, 14(8) (2022), 1596.
• [26] F.T. Okumuş, M. Akyiğit, K.J. Ansari and F. Usta, On approximation of Bernstein–Chlodowsky–Gadjiev type operators that fix $ e^{-2x} $, Adv. Contin. Discrete Models, 2022 2 (2022), 1-16.
• [27] M.A. Özarslan and O. Duman, Smoothness properties of modified Bernstein–Kantorovich operators, Numer. Funct. Anal. Optim., 37(1) (2016), 92–105.
• [28] P.J. Davis, Interpolation and Approximation. Dover Publications, Inc., New York, NY. (1975).
• [29] Z. Ditzian and V. Totik, Moduli of Smoothness, Springer, New York, (1987).
• [30] P.P. Korovkin, Linear Operators and Approximation Theory, Hindustan Publ. Corp., Delho, (1960).
• [31] R.A. Devore and G.G. Lorentz, Constructive Approximation, Springer: Berlin, Germany, 303 (1993).
• [32] J. Bustamante, Szasz–Mirakjan–Kantorovich operators reproducing affine functions, Results Math., 75(3),(2020), 1-13.
• [33] F. Dirik, and K. Demirci, Modified double Szasz-Mirakjan operators preserving $x^2+y^2$, Math. Commun., 15(1) (2010), 177-188.
• [34] J. Favard, Sur les multiplicateurs dinterpolation, Journal de Math´ematiques Pures et Appliqu´ees, 23(9) (1944), 219-47.
• [35] M. Örkcü, Szasz-Mirakyan-Kantorovich Operators of Functions of Two Variables in Polynomial Weighted Spaces, Abstr. Appl. Anal., 2013 (2013), 823803.
• [36] L. Rempulska and M. Skorupka, On Szasz-Mirakyan operators of functions of two variables, Matematiche, 53(1) (1998), 51-60.
• [37] V. Totik, Uniform approximation by Sz´asz-Mirakjan type operators, Acta Math. Hung., 41 (1983), 291-307.
• [38] 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 (N.S.), 115 (1957), 17-19.