\documentclass[draft,a4paper]{amsart}
\usepackage{amsmath,amsthm,amssymb,amsfonts}
\tolerance 1150
\newtheorem{othm}{Theorem}
\newcommand*{\cal}{\mathcal}
\newcommand*{\goth}{\mathfrak}
%\renewcommand{\baselinestretch}{0.992}
\DeclareMathOperator*{\infp}{inf\vphantom p}
\newcommand*{\wl}{\widehat\lambda}

\begin{document}

\begin{center}
{\large\bf Optimal recovery of linear operators}\\[5pt]
Osipenko K.Yu. (Moscow, Russia)
\end{center}


\vspace{10pt}

Let $X$ be a linear space, $Y_1,\ldots,Y_n$ be linear spaces with
semi-inner products, $I_j\colon X\to Y_j$, $j=1,\ldots,n$ be linear
operators, and $Z$ be a normed linear space. We consider the problem of
optimal recovery of a linear operator $T\colon X\to Z$ on a set
$$W=\{\,x\in X:\|I_jx\|_{Y_j}\le\delta_j,\ j=1,\ldots,k,\ k<n\,\}$$
by inaccurate information about values of the operators $I_{k+1},\ldots,I_n
$. More precisely, we are interesting in the value
$$E(T,W,I,\delta)=\infp_{\varphi\colon Y_{k+1}\times\ldots\times Y_n\to Z}
\,\,\,\sup_{\substack{x\in W,\ (y_{k+1},\ldots,y_n)\in Y_{k+1}\times\ldots
\times Y_n\\\|I_jx-y_j\|_{Y_j}\le\delta_j,\ j=k+1,\ldots,n}}\|Tx-\varphi(y)
\|_Z,$$
and in a method $\widehat\varphi$ for which this infimum is attained (we
call it an optimal method of recovery).

Consider the following extremal problem
\begin{equation}\label{O1}
\|Tx\|_Z^2\to\max,\quad\|I_jx\|_{Y_j}\le\delta_j^2,\ j=1,\ldots,n,\ x\in
X.
\end{equation}
Denote by
$$\mathcal L(x,\lambda)=-\|Tx\|_Z^2+\sum_{j=1}^n\lambda_j\|I_jx\|_{Y_j}^2$$
the Lagrange function of this problem.

\begin{othm}
Suppose that there exist nonnegative $\wl_j$, $j=1,\ldots,n$ such that
$\mathcal L(x,\wl)\ge0$ for all $x\in X$. Let $\{x_m\}$ be a sequence of
admissible elements in \eqref{O1} such that
\begin{align*}
(a)&\quad\lim_{m\to\infty}\mathcal L(x_m,\wl)=0,\\
(b)&\quad\lim_{m\to\infty}\sum_{j=1}^n\wl_j(\|I_jx_m\|_{Y_j}^2-\delta_j^2)=
0.
\end{align*}
If for all $y=(y_{k+1},\ldots,y_n)\in Y_{k+1}\times\ldots\times Y_n$ there
exists an element $x_y$ which is a solution of the problem
$$\sum_{j=1}^k\wl_j\|I_jx\|_{Y_j}^2+\sum_{j=k+1}^n\wl_j\|I_jx-y_j\|_{Y_j}^2
\to\min,\quad x\in X,$$
then the method
$$\widehat\varphi(y)=Tx_y$$
is optimal and
$$E(T,W,I,\delta)=\sqrt{\sum_{j=1}^n\wl_j\delta_j^2}.$$
\end{othm}

\vspace{5pt}
{\small This research was carried out with the financial support of the
Russian Foundation for Basic Research (grant nos. 02-01-39012 and
02--01--00386), 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 no.\ UR.04.03.067).}


\end{document}
