Some Approximation Results on $\lambda-$ Szasz-Mirakjan-Kantorovich Operators

Year 2021, Volume: 4 Issue: 3, 150 - 158, 30.09.2021

Reşat Aslan

https://doi.org/10.33401/fujma.903140

Cited By: 33

https://izlik.org/JA33XZ75HS

Abstract

In this article, we purpose to obtain several approximation properties of Sz\'{a}sz-Mirakjan-Kantorovich operators with shape parameter $\lambda \in\lbrack-1,1]$. We compute some preliminaries results such as moments and central moments for these operators. Next, we derive the Korovkin type convergence theorem, estimate the degree of convergence with respect to the moduli of continuity, for the functions belong to Lipschitz-type class and Peetre's $K$-functional, respectively. Further, we investigate Voronovskaya type asymptotic theorem and give the comparison of the convergence of these newly defined operators to the certain functions with some graphics.

Keywords

Degree of convergence , Lipschitz type functions , Moduli of continuity , $\lambda $-Sz\'{a}sz-Mirakjan polynomials , Voronovskaya type asymptotic theorem

References

• [1] O. Sz´asz, Generalization of the Bernstein polynomials to the infinite interval, J. Res. Nat. Bur. Stand., 45 (1950), 239-245. [2] G. M. Mirakjan, Approximation of continuous functions with the aid of polynomials, In Dokl. Acad. Nauk SSSR, 31 (1941), 201-205.
• [3] Z. Ditzian, V. Totik, Moduli of Smoothness, Springer, New York, 1987.
• [4] V. Gupta, R. P. Pant, Rate of convergence for the modified Sz´asz-Mirakyan operators on functions of bounded variation, J. Math. Anal. Appl., 233 (1999), 476-483.
• [5] N. ˙Ispir, Ç . Atakut, Approximation by modified Sz´asz-Mirakyan operators on weighted spaces, Proc. Math. Sci., 112 (2002), 571-578.
• [6] A. Aral, G. Ulusoy, E. Deniz, A new construction of Sz´asz-Mirakyan operators, Numer. Algorithms, 77 (2017), 313-326.
• [7] V. Totik, Uniform approximation by Sz´asz-Mirakian operators, Acta Math. Acad. Sci. Hungar., 41 (1983), 291-307.
• [8] S. G. Gal, Approximation with an arbitrary order by generalized Sz´asz-Mirakyan operators, Studia Univ. Babes-Bolyai Math., 59(1) (2014), 77-81.
• [9] D. Zhou, Weighted approximation by Sz´asz-Mirakian operators, J. Approx. Theory, 76 (1994), 393-402.
• [10] V. Gupta, V. Vasishtha, M. K. Gupta, Rate of convergence of the Sz´asz-Kantorovitch-Bezier operators for bounded variation functions, Publ. Inst. Math., (Beograd) (N.S.) 72 (2002), 137-143.
• [11] O. Duman, M. A. Özarslan, Sza´sz-Mirakjan type operators providing a better error estimation, Appl. Math. Lett., 20 (2007), 1184–1188.
• [12] O. Duman, M. A. Özarslan, B. D. Vecchia, Modified Sza´sz–Mirakyan–Kantorovich operators preserving linear functions, Turk J. Math., 33 (2009), 151–158.
• [13] Q. Qi, D. Guo, G. Yang, Approximation properties of l-Sz´asz-Mirakian operators, Int. J. Eng. Res., 12 (2019), 662-669.
• [14] Q.-B. Cai, B. Y. Lian, G. Zhou, Approximation properties of l-Bernstein operators, J. Inequal. Appl., 2018 (2018), 61.
• [15] Q.-B. Cai, G. Zhou, J. Li, Statistical approximation properties of l-Bernstein operators based on q􀀀integers, Open Math., 17 (2019), 487-498.
• [16] F. Özger, Applications of generalized weighted statistical convergence to approximation theorems for functions of one and two variables, Numer. Funct. Anal. Optim., 41 (16) (2020), 1990-2006.
• [17] F. Özger, Weighted statistical approximation properties of univariate and bivariate l-Kantorovich operators, Filomat, 33 (2019), 3473-3486.
• [18] F. Özger, On new Bezier bases with Schurer polynomials and corresponding results in approximation theory, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 69 (2020), 376-393.
• [19] H. M. Srivastava, F. Özger, S. A. Mohiuddine, Construction of Stancu-type Bernstein operators based on Bezier bases with shape parameter l, Symmetry, 11 (2019), 316.
• [20] M. Mursaleen, A. A. H. Al-Abied, M. A. Salman, Chlodowsky type (l;q)-Bernstein-Stancu operators, Azerb. J. Math., 10(1) (2020), 75-101.
• [21] A. M. Acu, N. Manav, D. F. Sofonea, Approximation properties of l-Kantorovich operators, J. Inequal. Appl., 2018 (2018), 202.
• [22] S. Rahman, M. Mursaleen, A. M. Acu, Approximation properties of l-Bernstein-Kantorovich operators with shifted knots, Math. Meth. Appl. Sci., 42 (2019), 4042-4053.
• [23] A. Kumar, Approximation properties of generalized l-Bernstein-Kantorovich type operators, Rend. Circ. Mat. Palermo (2), (2020), 1-16.
• [24] F. Özger, K. Demirci, S. Yıldız, Approximation by Kantorovich variant of l-Schurer operators and related numerical results, In: Topics in Contemporary Mathematical Analysis and Applications, pp. 77–94. CRC Press, Boca Raton (2020). ISBN 9780367532666
• [25] P. P. Korovkin, On convergence of linear positive operators in the space of continuous functions, Dokl. Akad. Nauk SSSR, 90 (1953), 961-964.
• [26] R. A. DeVore, G. G. Lorentz, Constructive Approximation, Springer, Heidelberg, 1993.
• [27] F. Altomare, M. Campiti, Korovkin-type approximation theory and its applications, volume 17, Walter de Gruyter, 2011.