Iterates of convolution-type operators

This paper is devoted to the study of the operators L having the property that there exists c∈[0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c\in [0,1)$$\end{document} with |Lf|Lip≤c|f|Lip\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|Lf|_{Lip}\le c|f|_{Lip}$$\end{document}. A class of convolution type operators acting on spaces of continuous periodic functions for which there exists the above mentioned constant c is described. We investigate the asymptotic behaviour of the iterates of such operators and give rates of convergence of the iterates toward the limit. Some examples involving the convolution operators of de la Vallée Poussin, Jackson and Féjer–Korovkin are presented. The well-known connection between de la Vallée Poussin operators and Durrmeyer operators with Chebyshev weights is used in order to obtain new estimates of the rates of convergence of the iterates of these operators.