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

The Zygmund sums of a function $f\in L_{1}$ are trigonometric polynomials the form $$Z^{s}_{n-1}(f;t):=\frac{a_{0}}{2}+\sum_{k=1}^{n-1}\Big(1-\big(\frac{k}{n}\big)^{s}\Big) \big(a_{k}(f)\cos kt+b_{k}(f)\sin kt\big), s>0,$$ where $a_{k}(f)$ and $b_{k}(f)$ Fourier coefficients $f$. We establish exact-order estimates uniform approximations by $Z^{s}_{n-1}$ $2\pi$-periodic continuous functions from classes $C^{\psi}_{\beta,p}$. These defined convolutions unit ball in space $L_{p}$, $1\leq p<\infty$, with generating fixed kernels $$\Psi_{\beta}(t)\sim\sum_{k=1}^{\infty}\psi(k)\cos\left(kt+\frac{\beta\pi}{2}\right), \Psi_{\beta}\in L_{p'}, \beta\in \mathbb{R}, \frac1p+\frac{1}{p'}=1.$$ additionally assume that product $\psi(k)k^{s+1/p}$ is generally monotonically increasing rate some power function, and, besides, for $1< p<\infty$ it holds $\sum_{k=n}^{\infty}\psi^{p'}(k)k^{p'-2}<\infty$, $p=1$ following condition $\sum_{k=n}^{\infty}\psi(k)<\infty$ true. It shown, under these conditions Fejér $\sigma_{n-1}=Z^{1}_{n-1}$ realize order best classes, namely $1<p<\infty$ $${E}_{n}(C^{\psi}_{\beta,p})_{C}\asymp{\cal E}\left(C^{\psi}_{\beta,p}; Z_{n-1}^{s}\right)_{C}\asymp\Big(\sum_{k=n}^{\infty}\psi^{p'}(k)k^{p'-2}\Big)^{1/p'}, \ \frac{1}{p}+\frac{1}{p'}=1,$$ $$ {E}_{n}(C^{\psi}_{\beta,1})_{C}\asymp{\cal E}\left(C^{\psi}_{\beta,1}; Z_{n-1}^{s}\right)_{C}\asymp {\left\{{\begin{array}{l l} \sum\limits_{k=n}^{\infty}\psi(k), & \cos \frac{\beta\pi}{2}\neq 0,\\ \psi(n)n, &\cos \frac{\beta\pi}{2}= 0, \end{array}} \right.} $${E}_{n}(C^{\psi}_{\beta,p})_{C}:=\sup_{f\in C^{\psi}_{\beta,p}}\inf\limits_{t_{n-1}\in\mathcal{T}_{2n-1}}\|f(\cdot)-t_{n-1}(\cdot)\|_{C}, $\mathcal{T}_{2n-1}$ subspace $t_{n-1}$ $n-1$ real coefficients, $${\cal Z_{n-1}^{s}\right)_{C}:=\mathop{\sup}\limits_{f\in C^{\psi}_{\beta,p}}\|f(\cdot)-Z^{s}_{n-1}(f;\cdot)\|_{C}.$$