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In this paper we recover a generalization of the classical Bernstein operators introduced by Morigi and Neamtu in 2000. Specifically, we focus on a sequence of operators that reproduce the exponential functions exp(μt) and exp(2μt) , μ > 0. We study its convergence, this including qualitative and quantitative theorems, an asymptotic formula and saturation results. We also show their shape preserving properties by considering generalized convexity. Finally, a comparison is stated, that shows that in a certain sense and for certain family of illustrative functions the new sequence approximates better than the classical Bernstein polynomials. Mathematics subject classification (2010): 41A36, 41A25, 41A40.

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In this paper, we present a generalization of the classical Korovkin theorem on positive linear operators. We deduce some convergence results for linear operators defined on C k [0, 1], that preserve some cones of functions related to shape properties. Finally, we show some examples. Academic Press