Korovkin type theorem and iterates of certain positive linear operators

TL;DR

This paper establishes a Korovkin type theorem for iterates of positive linear operators on continuous functions, providing convergence and approximation estimates, with applications to various q-operators.

Contribution

It extends Korovkin's theorem to iterates of general positive linear operators and offers quantitative convergence estimates.

Findings

Iterates of positive linear operators converge strongly to a fixed point.

Quantitative approximation estimates are derived using modulus of smoothness.

Results apply to several well-known q-operators.

Abstract

In this paper we prove Korovkin type theorem for iterates of general positive linear operators $T:C\left[ 0,1\right] \rightarrow C\left[ 0,1\right] $ and derive quantitative estimates in terms of modulus of smoothness. In particular, we show that under some natural conditions the iterates $T^{m}:C\left[ 0,1\right] \rightarrow C\left[ 0,1\right] $ converges strongly to a fixed point of the original operator $T$. The results can be applied to several well-known operators; we present here the $q$-MKZ operators, the $q$-Stancu operators, the genuine $q$-Bernstein--Durrmeyer operators and the Cesaro operators.