Problems of Information Transmission, Vol. 39, No. 1, 2003, pp. 104–118. Translated from Problemy Peredachi Informatsii, No. 1, 2003, pp. 118–133.
Original Russian Text Copyright
2003 by Magaril-Il’yaev, Osipenko, Tikhomirov.
Indeﬁnite Knowledge about an Object
and Accuracy of Its Recovery Methods
G. G. Magaril-Il’yaev
Department of Higher Mathematics, Moscow State Institute of Radio Engineering,
Electronics and Automation (Technology University)
Department of Higher Mathematics, Russian State Technological University (MATI)
Department of General Control Problems, Moscow State University
Abstract—An approach to the problem of optimal recovery of functionals and operators on
classes of functions under the conditions of inﬁnite knowledge of functions themselves is dis-
cussed. The capabilities of this approach are demonstrated in a number of examples. In the
end of the paper, a general result about optimal recovery of linear functionals is given.
Many aspects of human activity are connected with the fact that one has to judge on objects
under investigation using incomplete and/or inaccurate information about them. As a rule, it is
impossible to exactly recover an object from such information, so indeﬁniteness usually appears in
the form of a region where the object can be found. Sometimes, making the input information more
precise, we can approach the object closer and closer (in this case, one may consider the object as
“knowable”), but the “price” of such cognoscibility quite often turns out excessively high.
For a long time it was assumed that “world is knowable,” but now this is not persisted in since
fundamental bounds of cognoscibility were found (in mathematical logic, quantum mechanics, etc.).
On the other hand, if we have some information, we want to limit the indeﬁniteness as much as
possible, using this information to the maximum. For this purpose, some recovery methods are
applied to recover an object on the basis of available information. If a recovery method gives
bounds for an object that coincide with its measure of indeﬁniteness for the given information, we
may speak to the optimality of this recovery method.
A.N. Kolmogorov was interested in such problems during all his creative life and, in any case,
he faced them in his scientiﬁc activity (in probability theory, information theory, theory of ﬁring,
and many other problems). Some of the quantities introduced by him (for example, ε-capacity
and ε-entropy) are characteristics of measures of indeﬁniteness, and his results on extrapolation of
stochastic processes led to the corresponding optimal recovery methods.
In this paper, for a suﬃciently extensive class of problems (quite natural from the application
viewpoint), the notion of optimality of recovery methods from various types of information is
introduced. This approach is demonstrated by a number of examples, which have an illustrative
nature and are given to show a variety of problems covered by the proposed setting.
Supported in part by the Russian Foundation for Basic Research, project nos. 00-15-96109, 02-01-39012,
and 02-01-00386; program “Universities of Russia,” grant no. UR.04.03.013; and CRDF, grant no. VZ-
2003 MAIK “Nauka/Interperiodica”