Abstract
In the present article, we study certain approximation properties of the modified form of generalized Baskakov operators introduced by Erencin (Appl. Math. Comput. 218(3):4384–4390, 2011). We estimate a recurrence relation for the moments of their Durrmeyer type modification. First we estimate rate of convergence for functions having derivatives of bounded variation. Next, we discuss some direct results in simultaneous approximation by these operators, e.g. point-wise convergence theorem, Voronovskaja-type theorem and an estimate of error in terms of the modulus of continuity.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Acar, T., Gupta, V., Aral, A.: Rate of convergence for generalized Szász operators. Bull. Math. Sci. 1(1), 99–113 (2011)
Chen, W.Z.: Approximation Theory of Operators. Xiamen University Publishing House, Xiamen (1989)
Erencin, A.: Durrmeyer type modification of generalized Baskakov operators. Appl. Math. Comput. 218(3), 4384–4390 (2011)
Erencin, A., Bascanbaz-Tunca, G.: Approximation properties of a class of linear positive operators in weighted spaces. C. R. Acad. Bulgare Sci. 63(10), 1397–1404 (2010)
Goldberg, S., Meir, V.: Minimum moduli of ordinary differential operators. Proc. Lond. Math. Soc. 23(3), 1–15 (1971)
Gupta, V.: An estimate on the convergence of Baskakov–Bézier operators. J. Math. Anal. Appl. 312(1), 280–288 (2005)
Gupta, V.: A note on modified Baskakov type operators. Approx. Theory Appl. 10(3), 74–78 (1994)
Gupta, V., Agarwal, R.P.: Convergence Estimates in Approximation Theory. Springer, New York (2014)
Gupta, V., Srivastava, G.S.: Simultaneous approximation by Baskakov-Szász type operators. Bull. Math. de la Soc. Sci. de Roumanie (N. S.) 37 (85)(3–4), 73–85 (1993)
Hewitt, E., Stromberg, K.: Real and Abstract Analysis. McGraw Hill, New York (1956)
Ispir, N.: Rate of convergence of generalized rational type Baskakov operators. Math. Comput. Model. 46, 625–631 (2007)
Karsli, H.: Rate of convergence of new gamma type operators for functions with derivatives of bounded variation. Math. Comput. Model. 45, 617–624 (2007)
Mihesan, V.: Uniform approximation with positive linear operators generated by generalized Baskakov method. Autom. Comput. Appl. Math. 7(1), 34–37 (1998)
\(\ddot{O}\) zarslan, M.A., Duman, O., Kanna\(\check{g}\) lu, C.: Rates of convergence of certain King-type operators for functions with derivative of bounded variation. Math. Comput. Model. 52, 334–345 (2010)
Verma, D.K., Gupta, V., Agrawal, P.N.: Some approximation properties of Baskakov-Durrmeyer-Stancu operators. Appl. Math. Comput. 218(11), 6549–6556 (2012)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Gupta, V. (2016). Approximation for Generalization of Baskakov–Durrmeyer Operators. In: Rassias, T., Gupta, V. (eds) Mathematical Analysis, Approximation Theory and Their Applications. Springer Optimization and Its Applications, vol 111. Springer, Cham. https://doi.org/10.1007/978-3-319-31281-1_14
Download citation
DOI: https://doi.org/10.1007/978-3-319-31281-1_14
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-31279-8
Online ISBN: 978-3-319-31281-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)