On the best approximation of certain classes of periodic functions by trigonometric polynomials

TL;DR

This paper derives asymptotically exact estimates for the best uniform approximation of certain periodic function classes by trigonometric polynomials, based on their derivatives and modulus of continuity.

Contribution

It provides new asymptotic estimates for the best approximation of classes of periodic functions with specified derivative and continuity conditions.

Findings

Estimates are asymptotically exact under natural parameter conditions.

Results apply to classes defined by $(eta)$-derivatives and a majorant $oldsymbol{ extomega}$.

Enhances understanding of approximation accuracy for smooth periodic functions.

Abstract

We obtain the estimates for the best approximation in the uniform metric of the classes of $2\pi $-periodic functions whose $(\psi ,\beta)$-derivatives have a given majorant $\omega$ of the modulus of continuity. It is shown that the estimates obtained here are asymptotically exact under some natural conditions on the parameters $\psi ,$ $\omega $ and $\beta $ defining the classes