On approximating periodic functions by Riesz sums

Let $C$ be the space of continuous $2\pi$-periodic functions $f$ with the norm $\|f\|=\max_{x\in\mathbb R}|f(x)|$, $\sigma_n(f)$ be the Fejér sums of $f$, $E_n(f)$ be the best approximation, and let
\begin{align*} &X_n(f,a,x)=f(x)-\sigma_n(f,x)\\ &+\frac1{\pi(n+1)}\int^\infty_{a/(n+1)}\frac{f(x+t)+f(x-t)-2f(x)}{t^2}\,dt\\ &+\frac2\pi\sum^\infty_{k=1}\frac{(-1)^ka^{2k-1}\{(E-\sigma_n)^{2k}(f,x)\}}{(2k)!(2k-1)}. \end{align*}
Generalizing earlier results by M. Zamanskii and A. V. Efimov, the second author proved that for $f\in C$, the following relation is valid:
\begin{equation} \|X_n(f,a)\|\le C(a)E_n(f). \end{equation}
The present paper establishes advanced analogs of inequality (1) for the Riesz sums. Bibl. – 9 titles.