In this paper we introduce a new class of linear approximating operators \(\left( L_{nr}\right) _{n\geq1},r=0,1,2,…,\) for the functions \(f\in C^{r}\left[ 0,1\right]\). In order to construct them we use Taylor’s polynom of \(r\) degree and a classical class of linear positive operators generated by a probabilistic method. Also, we study approximation degree with the modulus of continuity of first and second order. \(\left( L_{nr}\right) _{n\geq1}\) include as a particular case the generalized Bernstein polynomials defined by G.H. Kirov in [5].


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



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O. Agratini, On a class of linear approximating operators, Mathematica Balkanica N.S., 11 (1997) nos. 3-4, 407-412.


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Mathematica Balkanica

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[1] O. Agratini, On the construction of approximating linear positive operators by probabilistic methods. Studia Univ. Babes-Bolyai 38, no.4 (1993), 45-50.
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[5] G.H. Kirov, A generalization of the Bernstein polynomials.  Mathematica Balkanica 6, no.2 (1992), 147-153.
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[7] D.D. Stancu, Use of probabilistic methods in the theory of uniform approximation of continuous funcitons.  Rev. Roum. Math.Pures Appl. 14, no.5 (1969, 673-691.
[8] D.D. Stancu, Probabilistic methods in the theory of approximaiton of functions of several variables by linear positive operators,  In: Approximation Theory (ed. A. Talbot), London-New York, 1970, 329-342.

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