A modulus of smoothness for some Banach function spaces

Abstract

UDC 517.5 Based on the Steklov operator, we consider a modulus of smoothness of functions in some Banach function spaces, which can be not translation invariant and establish its main properties.  A constructive characterization of the Lipschitz class is obtained by using the Jackson-type direct theorem and inverse theorem of trigonometric approximation. As application, we present several examples of related (weighted) function spaces.