In this paper we obtain estimations of the errors in approximation by positive linear operators which fix certain functions. We use both the first and the second order classical moduli of smoothness and a generalized modulus of continuity of order two. Some applications involving Bernstein type operators, Kantorovich type operators and genuine Bernstein-Durrmeyer type operators are presented.
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 D. Cardenas-Morales, P. Garrancho, I. Raşa, Bernstein-type operators which preserve polynomials, Comput. Math. Appl. 62, 2011, 158-163.
 W. Chen: On the modified Bernstein-Durrmeyer operators, Report of the Fifth Chinese Conference on Approximation Theory, Zhen Zhou, China, 1987.
 G. Freud, On approximation by positive linear methods I, II, Stud. Sci. Math. Hung. 2, 1967, 63-66; 3, 1968, 365-370.
 T.N.T. Goodman, A. Sharma, A modified Bernstein-Schoenberg operator, in: Proc. of the Conference on Constructive Theory of Functions, Varna 1987 (ed. by Sendov et al.), Sofia Publ. House Bulg. Acad. of Sci. 1988, 166-173.
 S. Karlin, W. Studden, Tchebyche Systems: with Applications in Analysis and Statistics, Interscience, New York, 1966.
 R. Paltanea, Optimal estimates with modulus of continuity, Result. Math. 32, 1997, 318-331.
 R. Paltanea, Estimates with adapted moduli of continuity for a Chebyshev system, Proc. of Int. Conf. Numerical Analysis and Approximation Theory (O. Agratini and P. Blaga Eds), Cluj-Napoca, July 4-8, 2006, 337-352.
 O. Shisha, B. Mond, The degree of convergence of sequences of linear positive operators, Proc. Nat. Acad. Sci. U.S.A. 60, 1968, 1196-1200.