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\begin{document}
\pagestyle{empty}
\title{Optimal recovery of functions from Hardy--Sobolev classes}
\author{K. Yu. Osipenko (Moscow)}

\maketitle

Let $X$ be a normed linear space of functions defined on some set $E$.
Suppose that $B$ is a subspace of $X$ and $l_1,\ldots,l_n\in B^*$. Set

\begin{align}\label{1}
i_n(B,X)&:=\infp_{l_1,\ldots,l_n\in B^*}\,\infp_{S\colon\mathbb R^n\to
X}\,\sup_{\|f\|_B\le1}\|f-S(l_1f,\ldots,l_nf)\|_X,\\
s_n(B,X)&:=\infp_{t_1,\ldots,t_n\in E}\,\infp_{S\colon\mathbb R^n\to
X}\,\sup_{\|f\|_B\le1}\|f-S(f(t_1),\ldots,f(t_n))\|_X.\notag
\end{align}
Any functionals for which the infimum in \eqref1 is attained we shall call
optimal functionals. The values $i_n(B,X)$ and $s_n(B,X)$ were introduced
by S.~Fisher and C.~Micchelli \cite1.

We study these quantities for the Hardy--Sobolev class $\HR$ which is the
set of all $2\pi$-periodic, real on the real axis and analytic in the strip
$S_\beta:=\{z:|\IM z|<\beta\}$ functions such that $|f^{(r)}(z)|\le1$,
$z\in S_\beta$. We show that $i_n(\HR,C)$ coincides with the Kolmogorov,
linear and Gel'fand $n$-widths of $\HR$ in $C$. Moreover, we prove that
Fourier coefficients $\{a_j(f)\}_{j=0}^k$, $\{b_j(f)\}_{j=1}^k$ are optimal
linear functionals in problem \thetag1 for $n=2k-1,2k$.

For $r=0$ and even $n$ we also prove that evaluations of $f\in\HH$ in the
system of equidistant points from $[0,2\pi)$ are optimal functionals, too.
That is
$$i_{2k}(\HH,C)=s_{2k}(\HH,C).$$
This equality does not valid for odd $n$. We show that
$$i_{2k-1}(\HH,C)<s_{2k-1}(\HH,C).$$


\begin{thebibliography}{9}
\bibitem{1} S.~D.~Fisher and C.~A.~Micchelli, Optimal sampling of holomorphic functions, II, {\it Math. Ann.\/} {\bf273} (1985), 131--147.
\bibitem{OW} K.~Yu.~ Osipenko, Exact $n$-widths of Hardy--Sobolev classes, {\it Constr. Approx.\/} (to appear).
\end{thebibliography}
\end{document}

