On a connection between a generalised modulus of smoothness of order~$r$ and the best approximation by algebraic polynomials

TL;DR

This paper introduces a new asymmetric operator of generalized translation to define a generalized modulus of smoothness, establishing direct and inverse approximation theorems for algebraic polynomial approximation.

Contribution

It presents a novel asymmetric operator of generalized translation and proves related approximation theorems, expanding the theoretical framework of polynomial approximation.

Findings

Established direct and inverse theorems for the generalized modulus of smoothness.

Connected the generalized modulus of smoothness with best polynomial approximation.

Extended approximation theory with a new operator and associated inequalities.

Abstract

In this paper an asymmetrical operator of generalised translation is introduced, the generalised modulus of smoothness is defined by its means and the direct and inverse theorems in approximation theory are proved for that modulus. ----- V danno\v{i} rabote vvoditsya nesimmetrichny\v{i} operator obobshchennogo sdviga, s ego pomoshchyu opredelyaetsya obobshchenny\v{i} modul' gladkosti i dlya nego dokazyvaetsya pryamaya i obratnaya teoremy teorii priblizheni\v{i}.