Approximation of periodic functions by Jackson type interpolation sums

Let
$$ \Phi_n(t)=\frac1{2\pi(n+1)}\Biggl(\frac{\sin\frac{(n+1)t}2}{\sin\frac t2}\Biggr)^2 $$
be Fejer's kernel, $C$ be a space of continuous $2\pi$-periodic functions $f$ with the norm $\|f\|=\max_{x\in\mathbb R}|f(x)|$;
$$ J_n(f,x)=\frac{2\pi}{n+1}\sum^n_{k=0}f(t_k)\Phi_n(x-t_k),\quad\text{where}\quad t_k=\frac{2\pi k}{n+1}, $$
be Jackson's polynomials of a function $f$, and let
$$ \sigma_n(f,x)=\int^\pi_{-\pi}f(x+t)\Phi_n(t)\,dt $$
be Fejer's sums of $f$.
The paper establishes upper estimates for the values of the types
$$ |f(x)-J_n(f,x)|,\quad|J_n(f,x)-\sigma_n(f,x)|,\quad\|f-J_n(f)\|,\quad\|J_n(f)-\sigma_n(f)\|, $$
which are exact in the order for every function $f\in C$. Special attention is paid to constants occurring in the inequalities obtained. Bibl. – 14 titles.