On some Bernstein type operators: iterates and generalizations


The present paper focuses on two approaches. Firstly, by using the contraction principle, we give a method for obtaining the limit of iterates of a class of linear positive operators. This general method is applied in studying three sequences of modified Bernstein type operators. Secondly, we define a generalization of Goodman-Sharma operators. We investigate the degree of approximation obtaining pointwise and global estimates in the framework of various function spaces.


Octavian Agratini
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania


linear positive operators; contraction principle; degree of approximation; modulus of smoothness.

Paper coordinates

O. Agratini, On some Bernstein type operators: iterates and generalizations, East Journal on Approximations, 9 (2003) no. 4, pp. 415-426.


About this paper


East Journal on Approximations

Publisher Name
Print ISSN


Online ISSN

google scholar link

[1] M. Campiti and G. Metafune, Approximation properties of recursively defined Bernstein-type operators,  J. Approx. Theory 87 (1996), pp. 243-269.
[2] E.W. Cheney and A. Sharma,  On a generalization of Bernstein polynomials,  Riv. Mat. Univ. Parma (2) 5 (1964), pp. 77-84.
[3] M.M. Derroemmoc,  Sur l’approximation de fonctions integrables sur [0,1] par des polynomes de Bernstein modifies,  J.Approx. Theory 31 (1981), pp.325-343.
[4] T.N.T. Goodman and A. Sharma,  A Bernstein-type operator on the simplex,  Math. Balkanica (N.S) 5 (1991), f.2, pp.129-145.
[5] R.P.Kelisky and T.J. Rivlin, Iterates of Bernstein polynomials, Pacific J. Math. 21 (1967(, pp. 511-520.
[6] I.A. Rus,  Weakly Picard mappings,  Commentationes Math. Univ. Carolinae 34, 4 (1993), pp. 769-773.
[7] D.D. Stancu, Approximation of functions by means of a new generalized Bernstein operator,  Calcolo, 20 (1983), f.2, pp.211-229.
[8] D.D. Stancu and C.Cismasiu,  On an approximating linear positive operator of Cheney-Sharma,  Revue d’Analyse Numerique et de Theorie de l’Approximation, 26 (1997), no.1-2, pp. 221-227.

Related Posts