Approximation rate and saturation under generalized convergence

Abstract

In this paper we prove a quantitative result about the convergence of sequences of functions defined from linear operators. The notion of convergence used here is the one given in [8]. The operators will be assumed to satisfy a shape preserving property associated with certain generalized derivative. We also study the saturation class, from the asymptotic condition that the sequence of operators fulfills. Finally, as applications, we show how the notion of weighted $ g $-statistical convergence, recently studied by A. Adem and M. Altinok [3], can be moved to the setting of approximation theory. Besides, we give a non standard example that shows the applicability of the results.