On moduli of smoothness and averaged differences of fractional order

TL;DR

This paper investigates fractional integral moduli of smoothness, establishing new equivalences with classical moduli and revealing some pathological effects specific to fractional cases.

Contribution

It introduces new equivalences between fractional and classical moduli of smoothness and discusses unique pathological effects in fractional integral settings.

Findings

New equivalences between fractional and classical moduli of smoothness

Identification of pathological effects in fractional integral moduli

Enhanced understanding of smoothness measures in approximation theory

Abstract

We consider two types of fractional integral moduli of smoothness, which are widely used in theory of functions and approximation theory. In particular, we obtain new equivalences between these moduli of smoothness and the classical moduli of smoothness. It turns out that for fractional integral moduli of smoothness some pathological effects arise.