Ovchintsev M.P. Optimal Recovery of Analytic Functions' Secondary Derivatives by Their Values at a Finite Number of Points

https://doi.org/10.15688/mpcm.jvolsu.2017.4.7

Mikhail  Petrovich  Ovchintsev
Candidate of Physical and Mathematical Sciences, Associate Professor,
Department of Applied Mathematics,
Moscow State University of Civil Engineering
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Yaroslavskoe shosse, 26, 129337 Moscow, Russian Federation

Abstract.  Many  works  are  devoted  to  the  problems  of  optimal  recovery  of  a  linear  functional
defined on a certain class of functions from information on the values of functions at a finite number of points (see, for example, [1; 2]). In this article we study the problem of best approximation of the second derivative of a bounded analytic function given in the unit circle at a point with respect to the value of the first derivative at this point, and also the values of the function in some finite collection of points. The article consists of three sections.
The  introduction  contains  the  necessary  information  from  the  articles  of  K.Yu.  Osipenko. The
definition of the best approximation method, the existence of the linear best method, and the formula for calculating the error of the best method are recalled. Also some necessary results from S.Ya. Khavinson articles are given.
In the second section, the error of the best approximation method is calculated. For this a family of
functions, which is used to find the error of the best method, is factorized. After this, the required error is calculated directly. It is noted, that the extremal function, used to determine the error of the best method, is unique up to a multiplier e, δR.
In the last section, the coefficients of the linear best method are calculated. To do this, we use the
corresponding contour integral, taken along the unit circle. This integral is estimated modulo from above, and then it is calculated. As a result, the coefficients of the linear best method are obtained. At the end of the paper, the uniqueness of the linear best method is established using the relation connecting the extremal functions (see [3]).

Key words: optimal recovery, analytic function, the best method, error of the best method, extremal
function, linear best method, coefficients of the linear best method.

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Optimal Recovery  of Analytic Functions' Secondary Derivatives by Their Values at a Finite Number  of  Points by Ovchintsev M.P. is licensed under a Creative Commons Attribution 4.0 International License.

Citation in English: Mathematical Physics and Computer Simulation . Vol. 20 No. 4 2017 pp. 76-82

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