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\centerline{\large\bf Some Problems of Optimal Recovery of Linear Operators}
%(pay attention to capital letters)
%example of too long name(if it does not fit on one line).the second line can be deleted
 \medskip
 \centerline{\bf Magaril-Il'yaev G.G.}
 \medskip
 \centerline{\it Faculty of Mechanics and Mathematics, Moscow State University}
 \centerline{\it magaril@mech.math.msu.su}
\medskip
 \centerline{\bf Osipenko K.Yu.}
 \medskip
 \centerline{\it Faculty of Mechanics and Mathematics, Moscow State University}
 \centerline{\it kosipenko@yahoo.com}


 \bigskip





In the report we will talk about the optimal recovery of functions from their
inexactly given spectrum and recovery of solutions of differential equations from inaccurate initial data. In the first case, as an illustration, we consider the optimal  recovery of
functions from inaccurate information about their Fourier coefficients. Next, for a
special one-parameter semi-group of operators, we consider the optimal recovery of the operator at a given value of the parameter from inaccurate information about the
values of other parameters. We construct a family of optimal methods. As a consequence we
find a family of optimal methods in the problem of optimal recovery for the solution
of the heat equation on $\mathbb R^d$
\begin{equation*}
\frac{\partial u}{\partial t}=\Delta u,\quad u(0,\cdot)=f(\cdot),
\end{equation*}
at the time instant $t$ from their approximate measurements at time instants $t_1<\ldots<t_n$. We also consider the problem of optimal recovery of the solution for the Dirichlet problem in the
half-space
\begin{equation*}
\Delta w\left(x, y\right)=0,\ \left(x,y\right)\in \mathbb R^{d}\times\mathbb R, \ y>0, \
w(\cdot, 0) = f(\cdot),
\end{equation*}
which is to recover the solution on the hyperplane $y=Y$ from its inaccurate measurements
on the hyperplanes $y=y_i$, $i=1,\ldots,n$.



\renewcommand\refname{\centering References}
\begin{thebibliography}{99}
\bibitem{1} {\it Magaril-Il'yaev G.G., Osipenko K.Yu.} On best harmonic synthesis of periodic functions //
J. Math. Sci. 2015. V. 209. N 1. P. 115-129.
\bibitem{2} {\it Magaril-Il'yaev G.G., Osipenko K.Yu.}  Optimal recovery of the solution of the heat
equation from inaccurate data // Sb. Math. 2009. V. 200. N 5. P. 665-682.
\end{thebibliography}

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