Functions measuring smoothness and the constants in Jackson--Stechkin theorem

TL;DR

This paper demonstrates that Jackson--Stechkin inequalities with optimal smoothness constants can be derived from Favard's inequalities, establishing an equivalence between two fundamental approximation theorems.

Contribution

It proves the equivalence of Favard's and Jackson--Stechkin inequalities, showing that the latter can be derived from the former with optimal constants.

Findings

Jackson--Stechkin inequality follows from Favard's inequality

Optimal smoothness constants are achieved in the derivation

Establishes equivalence of two key approximation theorems

Abstract

This paper is devoted to the equivalence of two type direct theorems in Approximation Theory: a) for smooth functions (Favard's estimates). b) for arbitrary continuous function (Jackson--Stechkin estimates). Specifically, we will show that Jackson--Stechkin inequality with optimal respect to the order of smoothness constants follows from Favard's inequality.