Stancu modified operators revisited


In this paper we construct a general positive approximation process representing an integral form in Kantorovich sense of the Stancu operators. By using K-functionals and some moduli of smoothness we give direct theorems for pointwise approximation. Also, by using the contraction principle we reobtain the convergence of the iterates of Stancu polynomials.


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


Kantorovich operators, Stancu operators, moduli of smoothness, K-functionals, contraction principle, weakly Picard operators.

Paper coordinates

O. Agratini, Stancu modified operators revisited, Revue d’Analyse Numerique et de Theorie de l’Approximation, 31 (2002) no. 1, 9-16.


About this paper


Revue d’Analyse Numerique et de Theorie de l’Approximation

Publisher Name

Publishing House of The Romanian Academy

Print ISSN


Online ISSN


google scholar link

[1] Agratini, O., On some properties by Stancu-Kantorovich polynomials in Lp spaces, in: Seminar of Numerical and Statistical Calculus (Gh. Coman, ed.), pp. 1–8, Babes-Bolyai Univ., Faculty of Math. and Computer Science, Cluj-Napoca, 1999.

[2] Altomare, F., and Campiti, M., Korovkin-Type Approximation Theory and its Applications, De Gruyter Series Studies in Mathematics,17, Walter de Gruyter, Berlin,1994.

[3] Della Vecchia, B. and Mache, D.H., On approximation properties of Stancu-Kantorovich operators, Rev. Anal. Numer. Theor. Approx.,27, no. 1, pp. 71–80, 1998.

[4] Ditzian, Z., and Totik, V., Moduli of Smoothness, Springer Series in ComputationalMathematics,9, Springer-Verlag, New York-Berlin, 1987.

[5] Guo, S., Liu, L., and X. Liu, The pointwise estimate for modified Bernstein operators, Studia Sci. Math. Hungarica,37, pp. 69–81, 2001.

[6] Lenze, B., On Lipschitz-type maximal functions and their smoothness spaces, Proc. Netherl. Acad. Sci. A,91, pp. 53–63, 1988.

[7] Mastroianni, G. and Occorsio, M.R., Una generalizatione del l’operatore di Stancu, Rend. Accad. Sci. Fis. Mat. Napoli, 45, no. 4, pp. 495–511, 1978.

[8] Q. Razi, Approximation of a function by Kantorovich type operators, Matematicki Vesnik,41, pp. 183–192, 1989.

[9] Rus, I.A., Iterates of Bernstein operators, via contraction principle, J. Math. Anal. Appl. (submitted).

[10] Stancu, D.D., Approximation of functions by a new class of linear polynomial operators, Rev. Roum. Math. Pures et Appl., 13, no. 8, pp. 1173–1194, 1968.

Related Posts