BEST METHODS FOR APPROXIMATING ANALYTIC FUNCTIONS GIVEN WITH AN ERROR

Автор: Osipenko K.Yu
Журнал: Mathematics of the USSR - Izvestiya
Том: 46
Номер: 3
Год издания: 1983
Первая страница: 353
Последняя страница: 374
DOI: 10.1070/SM1983v046n03ABEH002939
Аннотация: Let $B$ be the class of analytic functions of modulus at most 1 in the disk $\vert z\vert<1$, and let $z_1,\dots,z_n$ be distinct points in the interval $(-1, 1)$. This article takes up the problem of finding the quantity $\displaystyle r(z_0,z_1,\dots,z_n,\delta)=\inf_T\,\sup_{f\in B}\,\sup_{\Vert\widetilde{f}-\overline{f}\Vert_\infty\leq\delta}\vert f(z_0)-T(\widetilde{f})\vert,$ where the infimum is over all possible methods $T\colon\mathbf{R}^n\to\mathbf{R}$, $\widetilde{f}=(\widetilde{f}_1,\dots,\widetilde{f}_n)$, $\overline{f}=(f(z_1),\dots,f(z_n))$. It is determined that, depending on the error $\delta$, the information about the approximate values of functions in $B$ at some of the points can turn out to be superfluous. The order of informativeness of the system $z_1,\dots,z_n$ is found, i.e., the smallest $k$ for which there exists a subsystem $z_{i_1},\dots,z_{i_k}$ such that $r(z_0,z_{i_1},\dots,z_{i_k},\delta)=r(z_0,z_{1},\dots,z_{n},\delta)$. A best method of approximation is constructed, and the dependence of the order of informativeness on the size of the error $\delta$ is investigated. Bibliography: 21 titles.
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