Abstract
The concern of this paper is to introduce new generalized Durrmeyer-type operators from which classical operators can be obtained as a particular case, inspiring from the Ibragimov–Gadjiev operators (Gadjiev and Ibragimov, Soviet Math. Dokl. 11, 1092–1095, (1970) [8]). After the construction of new Durrmeyer operators is given, we obtain some pointwise convergence theorems and Voronovskaya-type asymptotic formula for new Durrmeyer-type operators. We establish a quantitative version of the Voronovskaya-type formula with the aid of the weighted modulus of continuity. Some special cases of new operators are presented as examples.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Aral, A.: Approximation by Ibragimov-Gadjiyev operators in polynomial weighted space. Proc. IMM NAS Azerbaijan XIX, 35–44 (2003)
Bardaro, C., Mantellini, I.: A quantitative Voronovskaya formula for Mellin convolution operators. Mediterr. J. Math. 7(4), 483–501 (2010)
Coskun, T.: On a construction of positive linear operators for approximation of continuous functions in the weighted spaces. J. Comp. Anal. Appl. 13(4), 756–770 (2011)
Derriennic, M.M.: Sur l’approximation de functions integrable sur [0; 1] par des polynomes de Bernstein modifies. J. Approx. Theory 31, 323–343 (1981)
Dogru, O.: On a certain family linear positive operators. Turk. J. Math. 21, 387–399 (1997)
Dogru, O., On the order of approximation of unbounded functions by the family of generalized linear positive operators. Commun. Fac. Sci. Univ. Ank., Ser. A1, 46, 173–181 (1997)
Durrmeyer, J.L.: Une formule d’ inversion de la Transformee de Laplace, Applications a la Theorie des Moments, These de 3e Cycle, Faculte des Sciences de l’ Universite deParis (1967)
Gadjiev, A.D., Ibragimov, I.I.: On a sequence of linear positive operators. Soviet Math. Dokl. 11, 1092–1095 (1970)
Gadjiev, A.D., İspir, N.: On a sequence of linear positive operators in weighted spaces. Proc. IMM Azerbaijan AS XI(XIX), 45–56 (1999)
Gonska, H., Pitul, P., Rasa, I.: On Peano’s form of the Taylor remainder,Voronovskaja’s theorem and the commutator of positive linear operators. In: Proceedings of the International Conference on Numerical Analysis and Approximation Theory, pp. 55–80. Cluj-Napoca, Romania, 5–8 July 2006
Gonska, H., Pitul, P., Rasa, I.: On differences of positive linear operators. Carpathian J. Math. 22(1–2), 65–78 (2006)
Gupta, V., Mohapatra, R.N., Finta, Z.: A certain family of mixed summation-integral type operators. Math. Comput. Model. 42(1–2), 181–191 (2005)
Heilmann, M.: Direct and converse results for operators of Baskakov-Durrmeyer type. Approx. Theory Appl. 5(1), 105–127 (1988)
Isir, N.: On modifed Baskakov operators on weighted spaces. Turk. J. Math. 25, 355–365 (2001)
Mazhar, S.M., Totik, V.: Approximation by modified Szasz operators. Acta Sci. Math. 49, 257–269 (1985)
Radatz, P., Wood, B., Approximating derivatives of functions unbounded on the positive axis with lineare operators, Rev. Roum. Math. Pures et Appl., Bucarest, Tome XXIII(5), 771–781 (1978)
Sahai, A., Prasad, G.: On simultaneous approximation bymodified Lupas operators. J. Approx. Theory 45(12), 122–128 (1985)
Srivastava, H.M., Gupta, V.: A certain family of summation integral type operators. Math. Comput. Model. 37(12–13), 1307–1315 (2003)
Voronovskaya, E.V.: Determination of the asymptotic form of approximation of functions by the polynomials of S.N. Bernstein, Dokl. Akad. Nauk SSSR, A, 79–85 (1932)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer Science+Business Media Singapore
About this paper
Cite this paper
Aral, A., Acar, T. (2016). On Approximation Properties of Generalized Durrmeyer Operators. In: Singh, V., Srivastava, H., Venturino, E., Resch, M., Gupta, V. (eds) Modern Mathematical Methods and High Performance Computing in Science and Technology. Springer Proceedings in Mathematics & Statistics, vol 171. Springer, Singapore. https://doi.org/10.1007/978-981-10-1454-3_1
Download citation
DOI: https://doi.org/10.1007/978-981-10-1454-3_1
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-10-1453-6
Online ISBN: 978-981-10-1454-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)