The rate of convergence of bounded linear processes on spaces of continuous functions

Abstract

Quantitative Korovkin-type theorems for approximation by bounded linear operators defined on \(C(X,d)\) are given, where \((X,d)\) is a compact metric space. Special emphasis is on positive linear operators. As is known from previous work of Newman and Shapiro, Jimenez Pozo, Nishishiraho and the author, among others, there are two possible ways to obtain error estimates for bounded linear operator approximation: the so-called direct approach, and the smoothing technique. We give various generalizations and refinements of earlier results which were obtained by using both techniques. Furthermore, it will be shown that, in a certain sense, none of the two methods is superior to the other one.

Keywords

• quantitative Korovkin-type theorems
• compact metric spaces
• bounded linear operators
• positive linear operators
• K-functional
• multivariate modulus of continuity