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\begin{document}

\title[Extremal problems and optimal recovery]{Extremal problems of the Hadamar three-circle theorem types and optimal recovery of operators}

\author{K.~Yu.~Osipenko}
\thanks{This research was carried out with the financial support of the
Russian Foundation for Basic Research (grant nos. 08-01-00450 and
08--01--90001)}
\address{MATI --- Russian State Technological 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
$$M(r)\le M(r_1)^{\frac{\log r_2/r}{\log r_2/r_1}}M(r_2)^{\frac{\log r/r_1}{\log r_2/r_1}}$$
for any three concentric circles of radii $r_1<r<r_2$.

In 1913 E.~Landau considered a very similar problem. He took derivatives instead of circles. He proved that for all functions $x\cd\in
\Lia$ with the first derivative locally absolutely continuous on $\mathbb
R_+$ and $x''\cd\in\Lia$ the following exact inequality
$$\|x'\cd\|_{\Lia}\le2\|x\cd\|_{\Lia}^{1/2}\|x''\cd\|_{\Lia}^{1/2}$$
holds. Then in 1914 Hadamard solved the analogous problem for $\mathbb R$.

In 1934 Hardy, Littlewood, and P\'olya proved that for all
integers $0<k<r$ the exact inequality
\begin{equation}\label{HL}
\|x^{(k)}\cd\|_{\lt}\le\|x\cd\|_{\lt}^{1-\frac kr}\|x^{(r)}\cd\|_{\lt}^{
\frac kr}
\end{equation}
holds for all functions $x\cd\in\lt$ for which the $(r-1)$-st derivative is locally absolute continuous on $\mathbb R$ and $x^{(r)}\cd\in\lt$.

The exact inequality \eqref{HL} may be easily obtained passing to the Fourier transforms from the following extremal problem
\begin{multline*}
\id\xi^{2k}|Fx(\xi)|^2\,d\xi\to\max,\quad\id|Fx(\xi)|^2\,d\xi\le\delta_1^2,\\
\id\xi^{2n}|Fx(\xi)|^2\,d\xi\le\delta_2^2,
\end{multline*}
where $Fx\cd$ is the Fourier transform of $x\cd$. The value of this extremal problem coincides with the value of the extremal problem
\begin{multline*}
\id\xi^{2k}|Fx(\xi)|^2\,d\xi\to\max,\quad\int_{|\xi|\le\sigma_1}|Fx(\xi)|^2\,d\xi
\le\delta_1^2,\\
\int_{|\xi|\ge\sigma_2}\xi^{2n}|Fx(\xi)|^2\,d\xi\le\delta_2^2
\end{multline*}
for some $\sigma_1\ge\sigma_2$. Using this fact we obtain a collection of optimal recovery methods of $x^{(k)}\cd$ from inaccurate information about the Fourier transform of $x\cd$.


\begin{thebibliography}{11}
\bibitem{MagOs} {\it Magaril-Il'yaev~G.~G., Osipenko~K.~Yu.}
Optimal recovery of functions and their derivatives from inaccurate
information about a spectrum and inequalities for derivatives, {\it Funkc.
analiz i ego prilozh.}, 37 (2003), 51--64; English transl. in {\it Funct.
Anal and Its Appl.}, 37 (2003), 203--214.
\end{thebibliography}


\end{document}