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\begin{document}
% begin contribution
\FAATrule
\title{Optimal Recovery of Analytic Functions}
\title{from Hardy--Sobolev Spaces}            %\label{filled by automatic processing}
                                         % WRITE THE TITLE OF YOUR ABSTRACT
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\author{K. Yu. Osipenko}%                             WRITE YOUR NAME INSIDE THE PRECEDING BRACKETS (CAPITALIZE ONLY INITIALS)
{MATI -- Russian State Technology University}%                              WRITE YOUR UNIVERSITY INSIDE THE PRECEDING BRACKETS (CAPITALIZE ONLY INITIALS)
{Russia}%                                 WRITE YOUR COUNTRY INSIDE THE PRECEDING BRACKETS (CAPITALIZE ONLY INITIALS)

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\bigskip\noindent\rm
{\bf AMS Classification: }{41A46, 30E05, 30D55}

\medskip\noindent
{\bf Keywords and phrases: }{Optimal recovery, Hardy--Sobolev spaces}
% \end Sections-organization

\bigskip
% \begin abstract

Denote by $H_{\infty,\beta}^r$ the class of $2\pi$-periodic, analytic in
the strip $S_\beta:=\{z\in\bbbc:|\Im z|<\beta\}$ functions $f$, which
satisfy the condition $|f^{(r)}(z)|\le1$, $z\in S_\beta$. We consider the
problem of optimal recovery of $f(\xi)$, $\xi\in\bbbt:=[0,2\pi)$, using the
information $If=(l_1f,\ldots,l_nf)$, where $l_jf$ are the Fourier
coefficients of $f$ or function values at a fixed system of nodes from
$\bbbt$.

We calculate the intrinsic error
$$e(\xi,H_{\infty,\beta}^r,I):=\infp_{S\colon\bbbc^n\to\bbbc}\,\sup_{f\in
H_{\infty,\beta}^r}|f(\xi)-S(If)|$$
and find an optimal algorithm of recovery $S_0$ for which the infimum is
attained. To obtain optimal recovery algorithms we use a method based on
parametrization of extremal functions in the dual extremal problem
$$\sup_\at{f\in H_{\infty,\beta}^r}{If=0}|f(\xi)|.$$


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