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

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.