| Abstract: |
| In this talk we will present some approximation results by means of a new class of polynomial-type operators that generalize the classical Bernstein polynomials and preserve a logarithmic function. The starting point are the Bernstein-type exponential polynomials introduced by Aral, Cardenas-Morales and Garrancho in 2018 and the main idea is to replace the exponential weights by means of logarithmic ones. For such operators we first establish pointwise and uniform convergence as well as their relation with the classical King operators. This allows us to obtain a quantitative estimate of the approximation error and a Voronovskaja-type asymptotic result. By means of this formula, we obtain saturation results and inverse theorems. In particular, by the Voronovskaja formula, a second order differential operator naturally arises: the set of the solutions of the corresponding homogeneous ODE represents the saturation class of the considered operators. Some shape preserving properties are also discussed. |
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