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\begin{center}  {\bf Optmal Recovery, Best Approximation, and Extremum
Theory}  \\
{G.~G.~Magaril-Il'yaev (Moscow State Institute of Radio Engineering,
Electronics and Automation), K.~Yu.~Osipenko (``MATI" --- Russian State
Technological University), V.~M.~Tikhomirov (Moscow State University)}
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The talk is devoted to the problems of optimal recovery of linear
functionals and operators by incomplete and/or inaccurate information and
their interconnections with the problems of analysis and approximation
theory. The last several years have seen the growing interest to the
problems where it is necessary to recover function values, or values of its
derivatives at some points, or value of integral of a function, or to
recover a function itself in some metric and so on. Usually in this
situation we have some information (incomplete and/or inaccurate) about
a function: for example, we know function values (may be with some errors)
at some points, or some of the Fourier coefficients, or moments, or initial
data if this function is a solution of differential equation and so on.
The problem is to use all this information in the best way for recovering
of one or another functional or operator from this function. The statement
of optimal recovery problem is going back conceptually to the papers of
Kolmogorov of the 30s of the past century where the problem of best
approximation method on a class of functions was stated.

In the talk a unique approach to investigation of optimal recovery
problems based on general principles of extremum theory and convex duality
is given. A series of examples will be presented where this approach allows
to obtain explicit expressions for optimal methods of recovery. In these
problems there are some interesting effects connected with excessive
information of input data which is apparently important in practical
applications. Moreover, it will be shown that there are close connections
between optimal recovery theory and problems of analysis and approximation
theory (polynomials least deviating from zero, problem of moments,
inequalities for derivatives of polynomials and smooth functions,
approximation of analytic functions).

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{\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))}
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