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<p>We consider the problems of optimal recovery of an operator <i>A</i> (generally speaking, nonlinear) defined on a unit ball <i>B</i><sub><i>H</i></sub> in a Hilbert space <i>H</i> based on the information about elements of this unit ball <i>B</i><sub><i>H</i></sub> given by a bounded linear operator <i>T</i> : <i>H</i> → <i>Y</i>, where <i>Y</i> is a Banach space. For a fixed information operator <i>T</i>, it is shown that the optimal method of recovery is offered by the so-called <i>T</i>-interpolating splines. For fixed <i>Y,</i> we also solve the problem of finding the optimal information operator. Moreover, for a bounded linear self-adjoint operator <i>A,</i> it is shown that if <i>T</i> is the optimal information operator for the recovery of <i>A</i> on <i>B</i><sub><i>H</i></sub><i>,</i> then any other operator <i>TA</i><sup><i>n</i></sup>, <i>n</i> ∈ ℕ, is also the optimal information operator.</p>

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Optimal Recovery of Mappings According to the Linear Data, Optimal Information Operators, and Extremal Subspaces

  • 期刊论文

  • 发表日期：

    2025年11月29日

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  • Vladyslav Babenko,
  • Yuliya Babenko,
  • Nataliia Parfinovych

摘要

We consider the problems of optimal recovery of an operator A (generally speaking, nonlinear) defined on a unit ball BH in a Hilbert space H based on the information about elements of this unit ball BH given by a bounded linear operator T : H → 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 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 BH, then any other operator TAn, n ∈ ℕ, is also the optimal information operator.