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\begin{document}
\pagestyle{empty}
\begin{center}\large\bf
ON OPTIMAL RECOVERY OF PERIODIC ANALYTIC FUNCTIONS
\end{center}
\begin{center}
\sc K. Yu. Osipenko, Moscow, Russia
\end{center}

Let $S_\beta:=\{z\in\bbbc:|\Im z|<\beta\}$ be a strip in the complex plane.
Denote by $\hp$ the class of $2\pi$-periodic, analytic in $S_\beta$
functions $f$, which satisfy
$$\displaylines{\sup_{0\le\eta<\beta}\frac1{4\pi}\int_{\bbbt}(|f(t+i\eta)|^
p+|f(t-i\eta)|^p)\,dt\le1,\quad1\le p<\infty,\cr
\sup_{z\in S_\beta}|f(z)|\le1,\quad p=\infty.}$$
We consider the problem of optimal recovery of $Lf=f(\xi)$ or $f'(\xi)$, $
\xi\in\bbbt$, using the information $If=(f(x_1),\ldots,f(x_n))$, $x_j\in
\bbbt$. We calculate the intrinsic error
$$e(L,\hp,I):=\infp_{A\colon\bbbc^n\to\bbbc\vphantom{\hp}}\,\sup_{f\in\hp}|
Lf-A(If)|\eqno(1)$$
and find an optimal algorithm $A^*$ for which the infimum in $(1)$ is
attained.

For example, if $Lf=f'(0)$ and $If=(f(-h),f(h))$, then an optimal algorithm
is given by
$$f'(0)\approx\frac K\pi\dn^{\frac{2(p-1)}p}\frac K\pi
h\,\,\frac{f(h)-f(-h)} {\sn\dfrac{2K}\pi h},$$
where $\dn z$ and $\sn z$ are the Jacobi elliptic functions with modulus $k
$ defined by the equation
$$\frac{\pi K'}{2K}=\beta$$
($K$ and $K'$ are the complete elliptic integrals of the first kind with
moduli $k$ and $k'=\sqrt{1-k^2}$).
\end{document}