Approximation of periodic functions in the uniform metric by Jackson type polynomials

Let $C$ be a space of continuous $2\pi$-periodic functions $f$ with the norm $\|f\|=\max_{x\in\mathbb R}|f(x)|$,
$$ J_n(f,x)=\frac1{(n+1)^2}\sum^n_{k=0}f(t_k)\Biggl(\frac{\sin\frac{(n+1)}2(x-t_k)}{\sin\frac{(x-t_k)}2}\Biggr)^2,\quad\text{where}\quad t_k=\frac{2\pi k}{n+1}, $$
be the Jackson polynomials of a function $f$. Let $\omega_r(f,h)$ be the $r$th continuity modulu of $f$, $E_n(f)$ be the best approximation of $f$ in the space $C$ by trigonometric polynomials of order $n$, and let $\widetilde F$ be the function trigonometrically conjugated with the primitive of $f$. The paper establishes results of the following types:
\begin{align*} E_n(f)+\|J_{4n-1}(f)-f\|&\approx\omega_1\Bigl(f,\frac1{n+1}\Bigr)+(n+1)\omega_2\Bigl(\widetilde F,\frac1{n+1}\Bigr),\\ \sup_{\alpha\in\mathbb R}\|J_n(f,\cdot+\alpha)-f(\cdot+\alpha)\|&\approx\omega_1\Bigl(f,\frac1{n+1}\Bigr)+(n+1)\omega_2\Bigl(\widetilde F,\frac1{n+1}\Bigr). \end{align*}
Here, the symbol $\approx$ does not depend on $f$ and $n$. Bibl. – 7 titles.