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

TL;DR

This paper derives precise estimates for how well Zygmund sums can uniformly approximate certain classes of periodic functions defined via convolutions, extending understanding of approximation rates for these function classes.

Contribution

It provides the exact order estimates of uniform approximation by Zygmund sums for classes of convolutions of periodic functions, under specific monotonicity and summability conditions.

Findings

Zygmund sums achieve the order of the best uniform approximation for these classes.

Conditions on the kernel functions ensure the approximation rates are optimal.

Results extend previous approximation theory for periodic function classes.

Abstract

We establish the exact-order estimates of uniform approximations by the Zygmund sums $Z^{s}_{n-1}$ of $2\pi$-periodic continuous functions $f$ from the classes $C^{\psi}_{\beta,p}$.These classes are defined by the convolutions of functions from the unit ball in the space $L_{p}$, $1\leq p<\infty$, with generating fixed kernels $\Psi_{\beta}(t)=\sum_{k=1}^{\infty}\psi(k)\cos\left(kt+\frac{\beta\pi}{2}\right)$, $\Psi_{\beta}\in L_{p'}$, $\beta\in \mathbb{R}$, $1/p+1/p'=1$. We additionally assume that the product $\psi(k)k^{s+1/p}$ is generally monotonically increasing with the rate of some power function, and, besides, for $1< p<\infty$ it holds that $\sum_{k=n}^{\infty}\psi^{p'}(k)k^{p'-2}<\infty$, and for $p=1$ the following condition is true $\sum_{k=n}^{\infty}\psi(k)<\infty$.It is shown that under these conditions Zygmund sums $Z^{s}_{n-1}$ and Fejer sums \linebreak$\sigma_{n-1}=Z^{1}_{n-1}$ realize the order of the best uniform approximations by trigonometric polynomials of these classes.