Korovkin type theorem for iterates of certain positive linear operators

TL;DR

This paper extends Korovkin-type theorems by proving convergence of iterates of certain positive linear operators on continuous functions, providing quantitative estimates based on smoothness measures.

Contribution

It introduces a broader class of positive linear operators for which the convergence of iterates is established with explicit quantitative estimates.

Findings

Iterates of the specified operators converge to a positive linear limit.

Quantitative convergence estimates are derived using moduli of smoothness.

The class of operators with known limit behavior is expanded.

Abstract

In this paper we prove that if T:C[0,1] \rightarrow C[0,1] is a positive linear operator with T(e_0)=1 and T(e_1)-e_1 does not change the sign, then the iterates T^{m} converges to some positive linear operator T^{\infty} :C[0,1] \rightarrow C[0,1] and we derive quantitative estimates in terms of modulii of smoothness. This result enlarges the class of operators for which the limit of the iterates can be computed and the quantitative estimates of iterates can be given.