Quantitative Korovkin theorems for sublinear, monotone and strongly translatable operators in $L^{p}([0, 1]), 1\le p \le +\infty$

TL;DR

This paper extends classical Korovkin theorems to sublinear, monotone, and strongly translatable operators in $L^{p}([0, 1])$, providing quantitative approximation estimates and exploring applications and open questions in interpolation theory.

Contribution

It introduces new quantitative Korovkin theorems for a broader class of operators, generalizing previous linear results to sublinear, monotone, and strongly translatable operators.

Findings

Established quantitative estimates using second and third order moduli of smoothness.

Extended classical Korovkin theorems to non-linear operator classes.

Included applications and posed open questions in interpolation theory.

Abstract

By extending the classical quantitative approximation results for positive and linear operators in $L^{p}([0, 1]), 1\le p \le +\infty$ of Berens and DeVore in 1978 and of Swetits and Wood in 1983 to the more general case of sublinear, monotone and strongly translatable operators, in this paper we obtain quantitative estimates in terms of the second order and third order moduli of smoothness, in Korovkin type theorems. Applications to concrete examples are included and an open question concerning interpolation theory for sublinear, monotone and strongly translatable operators is raised.