A special modulus of continuity and the K-functional

TL;DR

This paper introduces a new proof linking a special modulus of continuity with the K-functional and establishes Jackson's inequality for trigonometric polynomial approximation.

Contribution

It provides a novel proof of the equivalence between a special modulus of continuity and the K-functional, and proves Jackson's inequality for periodic functions.

Findings

Equivalence of Boman-Shapiro modulus and K-functional established

Jackson's inequality proved for trigonometric polynomial approximation

New proof techniques for approximation theory results

Abstract

We consider the questions connected with the approximation of a real continuous 1-periodic functions and give a new proof of the equivalence of the special Boman-Shapiro modulus of continuity with Peetre's K-functional. We also prove Jackson's inequality for the approximation by trigonometric polynomials.