Sharp estimates of approximation of periodic functions in H\"older spaces

TL;DR

This paper provides sharp estimates for approximating periodic functions in H"older spaces, improving classical theorems and establishing criteria for approximation rates across various parameters.

Contribution

It introduces improved direct and inverse theorems for approximation in H"older spaces using modified moduli of smoothness, and establishes strong converse inequalities.

Findings

Enhanced bounds for approximation in H"older spaces.

Criteria for the exact order of approximation decay.

Strong converse inequalities for general approximation methods.

Abstract

The main purpose of the paper is to study sharp estimates of approximation of periodic functions in the H\"older spaces $H_p^{r,\alpha}$ for all $0<p\le\infty$ and $0<\alpha\le r$. By using modifications of the classical moduli of smoothness, we give improvements of the direct and inverse theorems of approximation and prove the criteria for the precise order of decrease of the best approximation in these spaces. Moreover, we obtained strong converse inequalities for general methods of approximation of periodic functions in $H_p^{r,\alpha}$.