Abstract

We introduce a new class of positive linear operators that generalize the classical Bernstein operators. Specifically, we construct a sequence of operators that preserve the logarithmic function $\ln <!--FUNCTION\; APPLICATION-->(1+\mu +x)$, with $\mu >0$ and $x\in [0,1]$. We prove pointwise and uniform convergence and we derive a quantitative estimate of the approximation error in terms of the modulus of continuity. We also obtain a Voronovskaja-type asymptotic formula that is used to establish saturation results and inverse theorems. In particular, the saturation class of the considered approximation process is characterized by solving a second order differential equation. Shape-preserving properties, such as monotonicity, concavity and variation diminishing, are also investigated. Finally, a simple application to signal denoising is addressed.

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