A Kantorovich version of Bernstein-type logarithmic operators

TL;DR

This work introduces a Kantorovich variant of Bernstein-type logarithmic operators with logarithmic weights $\ln_\mu$ and analyzes their approximation properties. It proves pointwise and uniform convergence, establishes an $L^p$ convergence framework, and derives a Voronovskaja-type formula that yields a second-order differential operator governing the asymptotics. The saturation class is characterized by the corresponding homogeneous differential equation, and inverse theorems link the rate of approximation to this differential structure. Quantitative estimates are provided in both sup and $L^p$ norms via modulus of continuity and $K$-functionals, with explicit dependence on $\mu$ and sharp asymptotics for the associated auxiliary functions.

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.

Theorems & Definitions (18)

• Lemma 2.1
• Lemma 2.2
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 4.1
• proof
• Lemma 5.1
• Theorem 5.1