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\begin{document}

\title[Hadamard Type Extremal Problems]{Hadamard Type Extremal Problems and Optimal Recovery of Analytic Functions}

\author{K.~Yu.~Osipenko}
\address{Moscow State University}
\maketitle

The well-known Hadamard three-circle theorem states that if $f(z)$ is a holomorphic function on the annulus $r_1\le|z|\le r_2$ and
$$M(r)=\max_{|z|=r}|f(z)|,$$
then
$$M(\rho)\le M(r_1)^{\frac{\log r_2/r}{\log r_2/r_1}}M(r_2)^{\frac{\log r/r_1}{\log r_2/r_1}}$$
for any three concentric circles of radii $r_1<\rho<r_2$.

For functions $f$ from the Hardy space $\Hn$ we consider the analogous extremal problem
$$\|f(\rho z)\|_{\Hn}\to\max,\quad \|f(r_1z)\|_{\Hn}\le\delta_1,\quad \|f(r_2z)\|_{\Hn}\le\delta_2.$$
This problem is closely connected with the problem of optimal recovery of $f$ on the sphere of radius $\rho$ from the information about traces on the spheres of radii $r_1$ and $r_2$ given with errors. The optimal error of such recovery is defined as follows
\begin{multline*}
E_\rho(r_1,r_2,\delta_1,\delta_2)\\
=\inf_{m}\sup_{\substack{f\in\Hn,\ y_j\in L_2(\sigma_{r_j}),\ j=1,2\\
\|f(r_jz)-y_j(r_jz)\|_{\Ls}\le\delta_j,\ j=1,2}}\|f(\rho z)-m(y_1,y_2)(\rho z)\|_{\Ls},
\end{multline*}
where the lower bound is taken over all maps (methods) $m\colon L_2(\sigma_{r_1})\times L_2(\sigma_{r_2})\to L_2(\sigma_\rho)$ and 
$d\sigma_r(z)$ are the positive normalized rotationally invariant measures on the spheres $r\Sn$ ($\sigma=\sigma_1$). Any method $\wm$ for which the lower bound is attained is called an optimal recovery method.

Let
$$(\wl_1,\wl_2)=\left(\dfrac{r_2^2-\rho^2}{r_2^2-r_1^2}
\left(\dfrac\rho{r_1}\right)^{2s},\dfrac{\rho^2-r_1^2}{r_2^2-r_1^2}
\left(\dfrac\rho{r_2}\right)^{2s}\right),$$
if
$$\left(\frac{r_1}{r_2}\right)^{s+1}\le\frac{\delta_1}
{\delta_2}<\left(\frac{r_1}{r_2}\right)^s,\quad s\in\mathbb Z_+,$$
and $(\wl_1,\wl_2)=(0,1)$, if $\delta_1\ge\delta_2$.

\begin{theorem}[\cite{OS}]\label{RH}
The error of optimal recovery is given by
$$E_\rho(r_1,r_2,\delta_1,\delta_2)=\sqrt{\wl_1\delta_1^2+\wl_2\delta_2^2}$$
and the method
$$\wm(y_1,y_2)(z)=\sum_{k=0}^\infty\frac1{\wl_1r_1^{2k}+\wl_2r_2^{2k}}
\sum_{|\alpha|=k}(\wl_1r_1^kc_\alpha^{(1)}+\wl_2r_2^kc_\alpha^{(2)})z^\alpha,$$
where
%\begin{equation}\label{cc}
$$c_\alpha^{(j)}=\frac{(n+|\alpha|-1)!}{n!\alpha!}
\int_{\Sn}y_j(r_jz)\overline z^\alpha\,d\sigma(z),\quad j=1,2,$$
%\end{equation}
is optimal.
\end{theorem}

It appears that it is possible to construct a collection of optimal recovery methods.

\begin{theorem}\label{T2}
For all $\beta_k$, $k=0,1,\ldots$, such that
\begin{equation}\label{11}
\wl_2\left(\frac\rho{r_1}\right)^{2k}|\beta_k|^2+\wl_1
\left(\frac\rho{r_2}\right)^{2k}|1-\beta_k|^2\le\wl_1\wl_2
\end{equation}
all methods
$$\wm(y_1,y_2)(z)=\sum_{k=0}^\infty\sum_{|\alpha|=k}\left(\frac{\beta_k}{r_1^k}
c_\alpha^{(1)}+\frac{1-\beta_k}{r_2^k}c_\alpha^{(2)}\right)z^\alpha$$
are optimal.
\end{theorem}


Assume that $\delta_1<\delta_2$. Let $K_1=\max\{\,k\in\mathbb Z_+:\rho^{2k}\le\wl_1r_1^{2k}\,\}$, $K_2=\min\{\,k\in\mathbb Z_+:\rho^{2k}\le\wl_2r_2^{2k}\,\}.$

From Theorem~\ref{T2} we have

\begin{corollary}
For all $0\le k_1\le K_1$, $k_2\ge K_2$ and $\beta_k$, $k=k_1+1,\ldots,k_2-1$, such that \eqref{11} holds all methods
\begin{multline*}
m(y_1,y_2)(z)=\sum_{k=0}^{k_1}\sum_{|\alpha|=k}\frac{c_\alpha^{(1)}}{r_1^k}
z^\alpha\\
+\sum_{k=k_1+1}^{k_2-1}\sum_{|\alpha|=k}\left(\frac{\beta_k}{r_1^k}
c_\alpha^{(1)}+\frac{1-\beta_k}{r_2^k}c_\alpha^{(2)}\right)z^\alpha
+\sum_{k=k_2}^\infty\sum_{|\alpha|=k}\frac{c_\alpha^{(2)}}{r_2^k}
z^\alpha
\end{multline*}
are optimal.
\end{corollary}
\begin{thebibliography}{11}
\bibitem{OS} {\it Osipenko~K.~Yu., Stessin~M.~I.}
Hadamard and Schwarz type theorems and optimal recovery in spaces of analytic functions, {\it Constr. Approx.}, 31 (2010), 31--67.
\end{thebibliography}


\end{document}