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\begin{document}

\title{On Exact Values of $n$-Widths for Classes Defined by Nonlinear Cyclic Variation Diminishing Operators}
\author{K. Yu. Osipenko}

\maketitle

Denote by $h_\infty^\beta$ \ ($H_\infty^\beta$) the class of real-valued, $2\pi$-periodic functions which are analytic in the strip $S_\beta:=\{z\in\mathbb C:|\IM z|<\beta\}$ and satisfy the condition
$$|\RE f(z)|\le1\quad(|f(z)|\le1),\quad z\in S_\beta.$$


Let $P_n(D)$, $D=\dfrac d{dx}$, be a differential polynomial
with real coefficients. Denote by $h_\infty^{Q,\beta}$ \ ($H_\infty^{Q,
\beta}$) the class of real-valued, $2\pi$-periodic functions which are
analytic in $S_\beta$ and satisfy
$$Q(D)f\in h_\infty^\beta\quad(Q(D)f\in H_\infty^\beta),\quad z\in S_\beta.$$


Set
$$y(Q):=\max\{\,\IM z:Q(z)=0\,\}.$$
For the Kolmogorov ($d_n$), linear ($\delta_n$), and Gel'fand ($d_n$) $n
$-widths we prove that for all $n>2y(Q)$
\begin{multline*}
d_{2n}(W,C(\mathbb T))=\delta_{2n}(W,C(\mathbb T))=d^{2n}(W,C
(\mathbb T)
=d_{2n-1}(W,C(\mathbb T))\\=\delta_{2n-1}(W,C(\mathbb T))=d^{2n-1}(W,C(\mathbb T))=
\|\Omega_Q*\varphi(K_\beta*h_n)\|_\infty,
\end{multline*}
where
\begin{gather*}
\Omega_Q(t)=\sum_{\substack{k=-\infty\\Q(ik)\ne0}}^{+\infty}\frac{e^{ikt}}{Q(ik)},\quad
K_\beta(t)=1+2\sum_{k=1}^\infty\frac{\cos kt}{\ch k\beta},\\
h_n(t)=(-1)^{j+1},\quad\frac{(j-1)\pi}n\le t<\frac{j\pi}n,\quad
j=1,\ldots,2n,\\
\varphi(t)=\begin{cases}1,&W=h_\infty^{Q,\beta},\\
\tan\dfrac\pi4t,&W=H_\infty^{Q,\beta}.\end{cases}
\end{gather*}

To obtain this result we introduce special classes of functions defined by
cyclic variation diminishing operators which are not necessarily linear.

We also prove the analogous result for information $n$-widths
$$i_n(W,C(\mathbb T)):=\infp_{l_1,\ldots,l_n}\,\infp_{S\colon\mathbb R^n\to C(\mathbb T
)}\,\sup_{f\in W}\|f-S(l_1f,\ldots,l_nf)\|_\infty,$$
where $l_1,\ldots,l_n$ are any continuous linear functionals. Any
continuous linear functionals $l_1^*,\ldots,l_n^*$ for which the infimum is
attained are called optimal. We show that the first $2n-1$ Fourier
coefficients are optimal for $i_{2n}$ and $i_{2n-1}$ in the case $W=h_\infty^{Q,\beta}$ or $W=H_\infty^{Q,\beta}$.
\end{document} 