The paper focuses on the problem of optimal recovery of linear operators from inaccurately given values of other linear operators. Specifically, it considers the problem of recovering differential operators, such as powers of generalized Laplace operators and the Weil derivative, from a noisy Fourier transform.
The key highlights and insights are:
The paper establishes theoretical results on the error of optimal recovery and the construction of optimal recovery methods for these differential operators in the L2-metric.
For the case of powers of generalized Laplace operators, the paper provides explicit formulas for the error of optimal recovery and the optimal recovery methods. It also derives a sharp inequality relating the L2-norm of the recovered operator to the L2-norm of the Fourier transform and the L2-norm of the derivative.
For the case of the Weil derivative, the paper again provides explicit formulas for the error of optimal recovery and the optimal recovery methods, as well as a sharp inequality relating the L2-norm of the recovered derivative to the L2-norm of the Fourier transform and the L2-norm of another differential operator.
The results are obtained using techniques from optimal recovery theory, including the use of polar coordinates and homogeneous functions, as well as the application of Hölder's inequality.
Overall, the paper presents a comprehensive analysis of the optimal recovery of important classes of differential operators from noisy Fourier transform data, with explicit formulas and sharp inequalities that can be useful in various applications.
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by Konstantin Y... at arxiv.org 04-08-2024
https://arxiv.org/pdf/2404.03917.pdfDeeper Inquiries