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 $Z_{n-1}^s$ of $2\pi$-periodic continuous functions $f$ from the classes $C_{\beta ,p}^{\psi }$.These classes are defined by the convolutions of functions from the unit ball in the space $L_p$, $1\le p<\mathrm{\infty }$, with generating fixed kernels ${\mathrm{\Psi }}_{\beta }(t)={\sum }_{k=1}^{\mathrm{\infty }}\psi (k)\cos (kt+\frac{\beta \pi }{2})$, ${\mathrm{\Psi }}_{\beta }\in L_{p^{\mathrm{\prime }}}$, $\beta \in \mathbb{R}$, $1\mathrm{/}p+1\mathrm{/}{p}^{\mathrm{\prime }}=1$. We additionally assume that the product $\psi (k)k^{s+1\mathrm{/}p}$ is generally monotonically increasing with the rate of some power function, and, besides, for $1<p<\mathrm{\infty }$ it holds that ${\sum }_{k=n}^{\mathrm{\infty }}{\psi }^{p^{\mathrm{\prime }}}(k)k^{p^{\mathrm{\prime }}-2}<\mathrm{\infty }$, and for $p=1$ the following condition is true ${\sum }_{k=n}^{\mathrm{\infty }}\psi (k)<\mathrm{\infty }$.It is shown that under these conditions Zygmund sums $Z_{n-1}^s$ and Fejer sums \linebreak${\sigma }_{n-1}=Z_{n-1}^1$ realize the order of the best uniform approximations by trigonometric polynomials of these classes.