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\begin{center}\large\bf
ON OPTIMAL RECOVERY OF DERIVATIVES OF ANALYTIC FUNCTIONS
\end{center}
\begin{center}
\sc K. Yu. Osipenko (Moscow, Russia)
\end{center}

Let $W$ be a class of sufficiently smooth real-valued or
complex-valued functions. We study the problem of optimal recovery of the
$k$-th derivative of $f\in W$ at some fixed point $\xi$ on the basis of
information $If$ where $I$ is a linear operator from a linear space $X
\supset W$ into a normed linear space $Y$. In general this information can
be given with some error. The value
$$e_k(\xi,W,I,\delta):=\infp_{S\colon Y\to\bbbr(\bbbc)}\,\sup_\at{f\in W}{
\|y-If\|_Y\le\delta}|f^{(k)}(\xi)-S(y)|$$
is called the intrinsic error of optimal recovery. Any function $S$ for
which the infimum is attained is called an optimal method of recovery.

We consider this problem for the Hardy classes and Hardy--Sobolev classes
of functions defined in the unit disk or in the strip $\{z\in\bbbc:|\Im z|<
\beta\}$.

If $W$ is a convex and balanced set, then by the duality
$$e_k(\xi,W,I,\delta)=\sup_\at{f\in W}{\|If\|_Y\le\delta}|f^{(k)}(\xi)|.$$

For the Hardy class $H_\infty$ one of the first result relating to this
extremal problem was obtained by J.~Dieudonn\'e in 1931. He proved that
$$\sup_\at{|f(z)|\le1,\
|z|<1}{f(0)=0}|f'(\xi)|=\cases{1,&$|\xi|\le\sqrt2-1$,\cr
\dfrac{1+|\xi|^2}{4|\xi|(1-|\xi|^2)},&$|\xi|>\sqrt2-1$.}$$
That is in the case when $k=1$, $If=f(0)$, and $\delta=0$ the unit disk
divided in two parts in which extremal functions have different forms.

We study how this property transforms in general cases. We obtain the
intrinsic error and optimal algorithms for the periodic Hardy classes.

\medskip

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\small
This work was supported in part by RFBR Grants \#96-01-00325 and
\#96-15-96072.

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