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

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}.