Optimal recovery of mappings, optimal information operators, and extremal subspaces
DOI:
https://doi.org/10.3842/umzh.v77i7.8942Keywords:
Abstract splines, optimal recovery, optimal information operators, widths, extremal subspacesAbstract
UDC 517.5
We consider the problems of optimal recovery of an operator $A$ (generally speaking, nonlinear) defined on a unit ball $B_H$ of the Hilbert space $H$ based on the information about elements of this unit ball $B_H$ given by a linear bounded operator $T\colon H\to Y,$ where $Y$ is a Banach space. For a fixed information operator $T,$ it is shown that the optimal method of recovery is offered by the so-called $T$-interpolating splines. For a fixed $Y$ we also solve the problem of finding the optimal information operator. Moreover, for a bounded linear self-adjoint operator $A,$ it is shown that if $T$ is the optimal information operator for the recovery of $A$ on $B_H,$ then any other operator $TA^n,$ $n\in\mathbb N,$ is also an optimal information operator.
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