Best methods for approximating analytic functions given with an error

K. Yu. Osipenko

Sbornik. Mathematics, Tome 46 (1983) no. 3, pp. 353-374

Résumé

Let $B$ be the class of analytic functions of modulus at most 1 in the disk $|z|1$, 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 $$ r(z_0,z_1,\dots,z_n,\delta)=\inf_T\,\sup_{f \in B}\,\sup_{\|\widetilde f-\overline f\|_\infty\leqslant\delta}\vert f(z_0)-T(\widehat f)|, $$ 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.