English

Korovkin type theorem and iterates of certain positive linear operators

Functional Analysis 2010-12-07 v1

Abstract

In this paper we prove Korovkin type theorem for iterates of general positive linear operators T:C[0,1]C[0,1]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 Tm:C[0,1]C[0,1]T^{m}:C\left[ 0,1\right] \rightarrow C\left[ 0,1\right] converges strongly to a fixed point of the original operator TT. The results can be applied to several well-known operators; we present here the qq-MKZ operators, the qq-Stancu operators, the genuine qq-Bernstein--Durrmeyer operators and the Cesaro operators.

Keywords

Cite

@article{arxiv.1012.1187,
  title  = {Korovkin type theorem and iterates of certain positive linear operators},
  author = {N. I. Mahmudov},
  journal= {arXiv preprint arXiv:1012.1187},
  year   = {2010}
}
R2 v1 2026-06-21T16:54:05.508Z