A new class of positive linear operators preserving logarithmic functions

TL;DR

This paper introduces a new class of positive linear operators that extend Bernstein operators by preserving logarithmic functions, with proven convergence, error estimates, and applications in signal denoising.

Contribution

The paper develops a novel class of operators that reproduce logarithmic functions, providing convergence analysis, asymptotic formulas, and shape-preserving properties, extending classical approximation methods.

Findings

Operators reproduce logarithmic functions $ ext{ln}(1+ ext{μ}+x)$.

Proven pointwise and uniform convergence of the operators.

Application demonstrated in signal denoising.

Abstract

In this paper, we introduce a new class of positive linear operators that generalize the classical Bernstein operators. Specifically, we construct a sequence of operators that reproduce the logarithmic function $\ln(1+\mu+x)$, 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.