A Kantorovich version of Bernstein-type logarithmic operators

TL;DR

This paper develops a new Kantorovich variant of Bernstein-type logarithmic operators, analyzing their convergence, asymptotic behavior, and approximation properties with quantitative estimates.

Contribution

It introduces a novel Kantorovich version of Bernstein-type logarithmic operators and studies their convergence, asymptotics, and approximation estimates.

Findings

Established pointwise, uniform, and L^p convergence.

Derived a Voronovskaja-type asymptotic formula.

Provided quantitative approximation estimates using modulus of continuity and K-functionals.

Abstract

In this paper, we introduce a Kantorovich version of the Bernstein-type logarithmic operators. The idea comes from the wide literature concerning exponential polynomials that preserve exponential functions: here, the exponential weights are replaced by logarithmic ones and the corresponding operators preserve the logarithmic functions. The pointwise, the uniform and the $L^p$ convergence are first established. Then, a Voronovskaja-type asymptotic formula is derived: from it, a second-order differential operator naturally arises, allowing the characterization of the corresponding saturation class. Finally, quantitative estimates for the order of approximation are provided in the continuous case, in terms of the modulus of continuity, and, in the $L^p$ case, by means of suitable $K$-functionals.