Korovkin type theorem and iterates of certain positive linear operators

N. I. Mahmudov

paper

Korovkin type theorem and iterates of certain positive linear operators

arXiv:1012.1187

Abstract

In this paper we prove Korovkin type theorem for iterates of general positive linear operators $T : C [0, 1] \to C [0, 1]$ and derive quantitative estimates in terms of modulus of smoothness. In particular, we show that under some natural conditions the iterates $T^{m} : C [0, 1] \to C [0, 1]$ 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.