The order of approximation of continuous functions by certain linear means of their Fourier series

The exact order of the quantity
$$A_n(\mathfrak M)=\sup_{\Lambda\in\Lambda^*}\sup_{f\in\mathfrak M}\max_x|L_n(f;x;\Lambda)-f(x)|$$
is determined, where $\Lambda^*$ is the class of linear methods of summation of Fourier series $L_n(f;x;\Lambda)$, satisfying
$$(n+1)^{p-1}\sum_{k=0}^n|\Delta\lambda_k^{(n)}|^p\leqslant K^*,\quad p>1$$
and $\mathfrak M$ is either the set of continuous functions $H(\omega)$ or $C(F)$. In