On Jackson's theorem for the modulus of smoothness determined by a nonsymmetric generalised shift operator

TL;DR

This paper introduces a class of asymmetric generalized translation operators, defines corresponding moduli of smoothness, and proves Jackson's and its converse theorems for these moduli, extending classical approximation theory results.

Contribution

It extends Jackson's theorem to a new class of nonsymmetric generalized shift operators and their associated moduli of smoothness.

Findings

Proved Jackson's theorem for the new moduli

Established converse theorems for these moduli

Extended approximation theory to nonsymmetric operators

Abstract

In this paper a class of asymmetrical operators of generalised translation is introduced, for each of them generalised moduli of smoothness are introduced, and Jackson's and its converse theorems are proved for those moduli. ----- V eto\v{i} rabote rassmatrivaetsya klass sesimmetrichnykh operatorov obobshchenogo sdviga, dlya kazhdogo iz nikh vvoditsya obobshchennye moduli gladkosti i dlya nikh dokazybaetsya teorma Dzheksona i teorema, obratnaya e\v{i}.