On the Steklov Averages in Operator Cosine Function Framework

Abstract

The Steklov averages (Steklov or integral means) are used in approximation theory of functions in different aspects. This article concerns the Steklov averages by using the operator cosine function framework. The operator cosine function offers a counterpart of the translation operator, which forms the basic concept for the modulus of continuity and for some approximation processes as well. We will show that the operator cosine function concept allows to define very general Steklov averages in an abstract Banach space. The approximation properties of these generalized Steklov averages appear to be quite similar to the properties of the Steklov averages in trigonometric approximation.