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Let $L_{p}$, $1\le p\le\infty,$ and $C$ be the spaces of $2\pi$-periodic functions with standard norms $\|\cdot\|_{L_p}$ and $\|\cdot\|_C$, respectively. Further, let $W^r_{\beta,p},\ r>0,\ \beta\in\mathbb{R},\ 1\le p\le \infty,$ be classes of $2\pi$-periodic functions $f$ that can be represented in the form of convolution
$ f(x)=\frac{a_0}{2}+\frac{1}{\pi}\int\limits_{-\pi}^{\pi}\varphi(x-t) B_{r,\beta}(t)dt, \ \ \ a_0\in\mathbb R,\quad(1)$
with Weyl–Nagy kernels of the form $B_{r,\beta}(t)=\sum\limits_{k=1}^\infty k^{-r}\cos\left(kt-\frac{\beta\pi}2\right),\ $ of function $\varphi$ satisfying the condition
$ \varphi\in B_p^0=\big\{\varphi\in L_p: \|\varphi\|_{L_p}\le1,\ \int\limits_{-\pi}^{\pi}\varphi(t) dt=0\big\}. $
The classes $W^r_{{\beta},p}$ are called the Weyl–Nagy classes, and the function $\varphi$ in representation $(1)$ is called the $(r,\beta)$-derivative of the function $f$ in the Weyl–Nagy sense and denoted by $f^r_\beta$.
Let $f\in C$. By $\tilde{S}_{n-1}(f;x)$ we denote a trigonometric polynomial of degree $n-1$, that interpolates $f(x)$ at the equidistant nodes $x_{k}^{(n-1)}=2k\pi/(2n-1)$, $k\in\mathbb{Z}$, i.e., such that
$ \tilde{S}_{n-1}(f;x_{k}^{(n-1)})=f(x_{k}^{(n-1)}), \quad k\in\mathbb{Z}.$
Theorem 1. Let $r>2,$
$\beta\in\mathbb{R},\ $ $x\in\mathbb{R}$ and $\ n\in\mathbb{N}.$ The following estimate is true
$ \tilde{\cal E}_n(W^r_{\beta,1};x)\!=\!\sup\limits_{f\in W^r_{\beta,1}}\left|f(x)\!-\!\tilde S_{n-1}(f;x)\right|\!=\!\left|\sin\frac{(2n\!-\!1)x}2\right| n^{-r}\left(\frac2{\pi(1\!-\!e^{-r/n})}\!+\!\mathcal{O}(1)\delta_{r,n}\right), $
where $\mathcal{O}(1)$ is a quantity uniformly bounded in all analyzed parameters,
$ \delta_{r,n}= \left\{ \begin{array}{ll} \displaystyle1+\frac n{r(r-2)},&2< r\le n+1,\\ \displaystyle\frac r{n^2}e^{-r/n},&n+1\le r\le n^2,\\ e^{-r/n}%\ \mbox{або} \ \left(1+\frac1n\right)^{-r}, &r\ge n^2.\\ \end{array} \right. $
This work was partially supported by the VolkswagenStiftung project "From Modeling and Analysis to Approximation" and by the grant from the Simons Foundation (1290607, AS) and (1290607, IS).
