Order estimates of the uniform approximations by Zygmund sums on the classes of convolutions of periodic functions
We establish the exact-order estimates of uniform approximations by the Zygmund sums $Z^{s}_{n-1}$ of $2π$-periodic continuous functions $f$ from the classes $C^ψ_{β,p}$.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 $Ψ_β(t)=\sum_{k=1}^{\infty}ψ(k)\cos\left(kt+\frac{βπ}{2}\right)$, $Ψ_β\in L_{p'}$, $β\in \mathbb{R}$, $1/p+1/p'=1$. We additionally assume that the product $ψ(k)k^{s+1/p}$ is generally monotonically increasing with the rate of some power function, and, besides, for $1< p<\infty$ it holds that $\sum_{k=n}^{\infty}ψ^{p'}(k)k^{p'-2}<\infty$, and for $p=1$ the following condition is true $\sum_{k=n}^{\infty}ψ(k)<\infty$.It is shown that under these conditions Zygmund sums $Z^{s}_{n-1}$ and Fejer sums \linebreak$σ_{n-1}=Z^{1}_{n-1}$ realize the order of the best uniform approximations by trigonometric polynomials of these classes.