\documentclass[12pt]{article}
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\DeclareMathOperator{\IM}{Im}
\DeclareMathOperator{\ch}{ch}
\DeclareMathOperator{\thh}{th}
\DeclareMathOperator*{\infp}{inf\vphantom p}

\newcommand*{\Kk}{\mathcal K_{k,\delta}^{r,\beta}}
\newcommand{\cd}{(\cdot)}
\newcommand*{\lt}{L_2(\mathbb R)}
\newcommand*{\Ht}{\mathcal H_2^\beta}
\newcommand*{\Hr}{\mathcal H_2^{r,\beta}}
\newcommand*{\hr}{H_2^{r,\beta}}
\newcommand*{\wv}{\widehat\varphi}

\begin{document}
\begin{center}  {\bf Optmal Recovery of Derivatives\\
and Exact Inequalities in Hardy Spaces}  \\
{K.~Yu.~Osipenko (``MATI" --- Russian State Technological University)}
\end{center}

Denote by $\Ht$ the Hardy space of functions $f\cd$ analytic in the strip $
S_\beta=\{z\in\mathbb C:|\IM z|<\beta\}$ and satisfying the condition
$$\|f\cd\|_{\Ht}=\biggl(\sup_{0\le\eta<\beta}\frac12\int_{\mathbb R}(|f(t+i
\eta)|^2+|f(t-i\eta)|^2)\,dt\biggr)^{1/2}<\infty.$$
The Hardy-Sobolev space $\Hr$ is the space of functions $f\cd$ analytic in
the strip $S_\beta$ for which $f^{(r)}\cd\in\Ht$. The class $\hr$ is the
set of functions $f\cd\in\Hr$ for which $\|f^{(r)}\cd\|_{\Ht}\le1$.

We consider the problem of optimal recovery of $f^{(k)}\cd$ on $\mathbb R$
for functions $f\cd\in\hr\cap\lt$, $k\le r$, by the information about the
trace of $f\cd$ on $\mathbb R$ given with some error. Put
$$E_k(\hr,\delta)=\infp_{\vphantom{\hr}\varphi\colon\lt\to\lt}\,\sup_{
\substack{f\cd\in \hr\cap\lt,\ y\cd\in\lt\\\|f\cd-y\cd\|_{\lt}\le\delta}}\|
f^{(k)}\cd-\varphi(y)\cd\|_{\lt}.$$
Any method $\wv$ for which this infimum is attained is called optimal.

Denote by $\mu_{r\beta}(x)$ the unique solution of the equation $t^r\sqrt{
\ch2\beta t}=x$ which belongs to the interval $[0,+\infty)$.

{\bf Theorem.} {\it For all $r,k\in\mathbb N$, $k\le r$, and $\delta>0$
$$E_k(\hr,\delta)=\sup_{\substack{f\cd\in\hr\cap\lt\\\|f\cd\|_{\lt}\le
\delta}}\|f^{(k)}\cd\|_{\lt}=\delta\mu_{r\beta}^k(\delta^{-1}).$$
Moreover, the method
$$\wv(y)\cd=(\Kk*y)\cd,$$
where
$$\Kk(x)
=\frac1{2\pi}\int_{\mathbb R}(it)^k\left(1+\frac{k\delta^2t^{2r}\ch2\beta t
}{r-k+\beta\mu_{r\beta}(\delta^{-1})\thh(2\beta\mu_{r\beta}(\delta^{-1}))}
\right)^{-1}e^{ixt}\,dt,$$
is optimal.}
\smallskip

{\small The research was carried out with the financial support of the
Russian Foundation for Basic Research (grant nos.\ 05-01-00275 and
05-01-00261), the President Grant for State Support of Leading Scientific
Schools in Russian Federation (grant no.\ NSH-304.2003.1), and the Program
``Universities of Russia" (grant nos.\ UR.03.01.130 and UR.04.02.536).}

\end{document}
