Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter December 9, 2015

On Approximation Properties of a New Type of Bernstein-Durrmeyer Operators

  • Tuncer Acar EMAIL logo , Ali Aral and Vijay Gupta
From the journal Mathematica Slovaca


The present paper deals with a new type of Bernstein-Durrmeyer operators on mobile interval. First, we represent the operators in terms of hypergeometric series. We also establish local and global approximation results for these operators in terms of modulus of continuity. We obtain an asymptotic formula for these operators and in the last section we present better error estimation for the operators using King type approach


[1] ABEL, U.-IVAN, M.-LI, Z.: Local approximation by generalized Baskakov-Durrmeyer operators, Numer. Funct. Anal. Optimiz. 28 (2007), 245-264.10.1080/01630560701277823Search in Google Scholar

[2] AGRATINI, O.: Linear operators that preserve some test functions, Int. J. Math. Math. Sci. (2006), Art. ID 94136, 11 pp..Search in Google Scholar

[3] AGRATINI, O.: On the iterates of a class of summation type linear positive operators, Comput. Math. Appl. 55 (2008), 3795-3801.10.1016/j.camwa.2007.04.044Search in Google Scholar

[4] ARAL, A.-GUPTA, V.: Generalized q Baskakov operators, Math. Slovaca 61 (2011), 619-634.10.2478/s12175-011-0032-3Search in Google Scholar

[5] DEVORE, R. A.-LORENTZ, G. G.: Constructive Approximation. Grundlehren Math. Wiss. 303, Springer-Verlag, Berlin-Heidelberg-New York-London, 1993.10.1007/978-3-662-02888-9_10Search in Google Scholar

[6] DERRIENNIC, M. M.: Sur l’approximation des fonctions integrables sur [0, 1] par des polynomes de Bernstein modifi´es, J. Approx. Theory 31 (1981), 325-343.10.1016/0021-9045(81)90101-5Search in Google Scholar

[7] DITZIAN, Z.-TOTOK, V.: Moduli of Smoothness, Springer-Verlag, New York, 1987.10.1007/978-1-4612-4778-4Search in Google Scholar

[8] DUMAN, O.-OZARSLAN, M. A.: Sz´asz-Mirakjan type operators providing a better error estimation, Appl. Math. Lett. 20 (2007), 1184-1188.10.1016/j.aml.2006.10.007Search in Google Scholar

[9] DURRMEYER, J. L.: Une formule d’inversion de la Transformee Laplace. Applications a la Theorie des Moments, These de 3e Cycle, Faculte des Sciences de I’ Universite de Paris, 1967.Search in Google Scholar

[10] GADJIEV, A. D.-GHORBANALIZADEH, A. M.: Approximation properties of a new type Bernstein-Stancu polynomials of one and two variables, Appl. Math. Comput. 216 (2010), 890-901.10.1016/j.amc.2010.01.099Search in Google Scholar

[11] GUPTA, V.: A note on the rate of convergence of Durrmeyer type operators for function of bounded variation, Soochow J. Math. 23 (1997), 115-118.Search in Google Scholar

[12] GUPTA, V.-MAHESHWARI, P.: Bézier variant of a new Durrmeyer type operators, Riv. Mat. Univ. Parma 7 (2003), 9-21.Search in Google Scholar

[13] GUPTA, V.-HEPING, W.: The rate of convergence of q-Durrmeyer operators for 0 < q < 1, Math. Methods Appl. Sci. 31 (2008), 1946-1955.10.1002/mma.1012Search in Google Scholar

[14] GUPTA, V.-YADAV, R.: Approximation by complex summation-integral type operators in compact disks, Mathematika 63 (2013), 1025-1036.Search in Google Scholar

[15] KING, J. P.: Positive linear operators which preserves x2, Acta Math. Hungar. 99 (2003), 203-208.10.1023/A:1024571126455Search in Google Scholar

[16] ÖZARSLAN, M. A.-DUMAN, O.: MKZ type operators providing a better estimation on [1/2, 1), Canad. Math. Bull. 50 (2007), 434-439.10.4153/CMB-2007-042-8Search in Google Scholar

[17] REMPULSA, L.-TOMCZAK, K.: Approximation by certain linear operators preserving x2, Turkish J. Math. 33 (2009), 273-281.Search in Google Scholar

[18] STANCU, D. D.: Approximation of functions by a new class of linear polynomial operators, Rev. Roumaine Math. Pures Appl. 13 (1968), 1173-1194.Search in Google Scholar

[19] ZENG, Z. M.-ABEL, U.-IVAN, M.: Approximation by a Kantorovich variant of the Bleimann Butzer and Hahn operators, Math. Inequal. Appl. 11 (2008), 317-325.Search in Google Scholar

[20] GUPTA, V.-ZENG, X. M.: On the Bézier variant of Balazs Kantorovitch operators, Math. Slovaca 57 (2007), 349-358. 10.2478/s12175-007-0029-0Search in Google Scholar

Received: 2012-8-22
Accepted: 2012-7-12
Published Online: 2015-12-9
Published in Print: 2015-10-1

Mathematical Institute Slovak Academy of Sciences

Downloaded on 31.3.2023 from
Scroll to top button