\documentclass[12pt,draft,a4paper]{amsart}
\usepackage{amsmath,amsthm}
\usepackage[T2A]{fontenc}
\usepackage[cp1251]{inputenc}
\usepackage[english]{babel}
\usepackage{amsfonts}
\usepackage{latexsym}
%\usepackage{srctex}
\tolerance 2000

\newtheorem{theorem}{Theorem}


\newcommand{\HH}{H_{2,\beta}}
\newcommand{\CC}{C(\mathbb T)}
\newcommand{\HC}{(\HH,\CC)}

\DeclareMathOperator*{\IM}{Im}
\DeclareMathOperator*{\infp}{inf\vphantom p}

\begin{document}
\pagestyle{empty}
\title{Optimal recovery of functions in $\HH$}
\author{K. Yu. Osipenko (Moscow, Russia)}

\maketitle

Let $S_\beta:=\{z\in\mathbb C:|\IM z|<\beta\}$ be a strip in the complex plane.
Denote by $\HH$ the class of $2\pi$-periodic, analytic in $S_\beta$ functions $f$, which satisfy
$$\sup_{0\le\eta<\beta}\frac1{4\pi}\int_0^{2\pi}(|f(t+i\eta)|^2+|f(t-
i\eta)|^2)\,dt\le1.$$
Set
\begin{multline*}
s_n\HC\\
:=\infp_{t_1,\ldots,t_n\in\mathbb T}\,\infp_{A\colon\mathbb C^n
\to\CC}\,\sup_{f\in\HH}\|f-A(f(t_1),\ldots,f(t_n))\|_{\CC},\\
i_n\HC:=\infp_{l_1,\ldots,l_n}\,\infp_{A\colon\mathbb C^n\to\CC}\,\sup_{f\in
\HH}\|f-A(l_1f,\ldots,l_nf)\|_{\CC},
\end{multline*}
where $l_1,\ldots,l_n$ are linear continuous functionals. The values $i_n\HC$ are coincide with linear and Gel'fand $n$-widths which were calculated
in \cite{Os}. We study the values $s_n\HC$ and compare them with $i_n\HC$.

Denote by $K$ and $K'$ be the complete elliptic integrals of the first kind
with moduli $k$ and $k'=\sqrt{1-k^2}$. Suppose that $k$ is defined by the
equation
$$\frac{\pi K'}{2K}=\beta.$$
We prove that
\begin{align*}
\frac{s_{2n-1}\HC}{i_{2n-1}\HC}&=2\sqrt{\frac{kK}\pi\sinh\beta}
\,+O(e^{-4\beta n}),\\
\frac{s_{2n}\HC}{i_{2n}\HC}&=2\sqrt{\frac K\pi\tanh\beta}\,+O(e^{-4\beta n}
).
\end{align*}

\begin{thebibliography}{9}
\bibitem{Os}
{\sc K. Yu. Osipenko.} On $n$-widths of holomorphic functions of several
variables. J. Approx. Theory {\bf 82}, 1995, 135--155.
\bibitem{OW}
{\sc K. Yu. Osipenko and K. Wilderotter}. Optimal information for
approximating periodic analytic functions. Math. Comput. (to appear).
\end{thebibliography}
\end{document}