Estimates of the uniform approximations by Zygmund sums on the classes of convolutions of periodic functions

TL;DR

This paper derives precise estimates for uniform approximation errors of certain classes of periodic functions using Zygmund sums, identifying parameter ranges where these sums achieve optimal approximation order.

Contribution

It provides order-exact uniform approximation estimates by Zygmund sums for classes of convolutions of periodic functions, including parameter conditions for optimality.

Findings

Order-exact uniform approximation estimates obtained.

Parameter ranges identified where Zygmund sums match best approximation order.

Results applicable to classes of functions representable by convolutions with fixed kernels.

Abstract

We obtain order-exact estimates for uniform approximations by using Zygmund sums $Z^{s}_{n}$ of classes $C^{\psi}_{\beta,p}$ of $2\pi$-periodic continuous functions $f$ representable by convolutions of functions from unit balls of the space $L_{p}$, $1< p<\infty$, with a fixed kernels $\Psi_{\beta}\in L_{p'}$, $\frac{1}{p}+\frac{1}{p'}=1$. In addition, we find a set of allowed values of parameters (that define the class $C^{\psi}_{\beta,p}$ and the linear method $Z^{s}_{n}$) for which Zygmund sums and Fejer sums realize the order of the best uniform approximations by trigonometric polynomials of those classes.