Complete Asymptotic Expansion for Generalized Favard Operators

In the present paper we consider a generalization \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$F_{n,\sigma_{n}} $\end{document} of the Favard operators and study the local rate of convergence for smooth functions. As a main result we derive the complete asymptotic expansion for the sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$( F_{n,\sigma _{n}}f)( x)$\end{document} as n tends to infinity. Furthermore, we consider a truncated version of these operators. Finally, all results were proved for simultaneous approximation.