Abstract: The paper considers a one-parameter family of linear continuous operators in \(L_2 (\mathbb{R ^d})\) and poses the problem of optimal reconstruction of the operator for a given value of a parameter on a class of functions whose Fourier transforms are square integrable with power weight (spaces of such a structure play an important role in questions of embedding function spaces and the theory of differential equations) using the following information: about each function from this class is knows (generally speaking, approximately) its Fourier transform on some measurable subset of \(\mathbb{R ^d}\). A family of optimal methods for restoring operators for each parameter value is constructed. Optimal methods do not use all the available information about the Fourier transform of functions from the class, but use only information about the Fourier transform of a function in a ball centered at zero of maximum radius, which has the property that its measure is equal to the measure of its intersection with the set, where is known (exactly or approximately) Fourier transform of the function. As a consequence, the following results were obtained: a family of optimal methods for recovery the solution of the heat equation in \(\mathbb{R ^d}\) at a given time, provided that the initial function belongs to the specified class and its Fourier transform is known exactly or approximately on some measurable set, and also a family of optimal methods for reconstructing the solution of the Dirichlet problem for a half-space on a hyperplane from the Fourier transform of a boundary function belonging to the specified class, which is known exactly or approximately on some measurable set in \(\mathbb{R ^d}\).
For citation: Abramova, E. V. and Sivkova, E. O. On the Best Recovery of a Family of Operators on a Class of Functions According to Their Inaccurately Specified Spectrum, Vladikavkaz Math. J., 2024, vol. 26, no. 1, pp.13-26 (in Russian). DOI 10.46698/z4058-1920-7739-f
1. Stein, E. M. and Weiss, G. Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 1971.
2. Micchelli, C. A. and Rivlin, T. J. A Survey of Optimal Recovery,
Optimal Estimation in Approximation Theory, Eds. C. A. Micchelli and T. J. Rivlin, New York, Plenum Press, 1977, pp. 1-54. DOI: 10.1007/978-1-4684-2388-4_1.
3. Melkman, A. A. and Micchelli, C. A. Optimal Estimation of Linear Operators in Hilbert Spaces From Inaccurate Data, SIAM Journal on Numerical Analysis, 1979, vol. 16, no. 1, pp. 87-105.
4. Micchelli, C. A. and Rivlin, T. J. Lectures on Optimal Recovery, Lecture Notes in Mathematics, Berlin, Springer-Verlag, 1985, vol. 1129, pp. 21-93.
DOI: 10.1007/bfb0075157.
5. Traub, J. F. and Wozniakowski, H. A General Theory of Optimal Algorithms, New York, Academic Press, 1980.
6. Magaril-Il'yaev, G. G. and Osipenko, K. Yu. Optimal Recovery of Functions and Their Derivatives from Inaccurate Information about the Spectrum and Inequalities for Derivatives, Functional Analysis and Its Applications, 2003, vol. 37, no. 3, pp. 203-214. DOI: 10.1023/A:1026084617039.
7. Magaril-Il'yaev, G. G. and Osipenko, K. Yu. On Optimal Harmonic Synthesis from Inaccurate Spectral Data, Functional Analysis and Its Applications, 2010, vol. 44, no. 3, pp. 223-225.
DOI: 10.1007/s10688-010-0029-7.
8. Magaril-Il'yaev, G. G. and Osipenko, K. Yu. On the Reconstruction of Convolution-Type Operators from Inaccurate Information, Proceedings of the Steklov Institute of Mathematics, 2010, vol 269, pp. 174-185. DOI: 10.1134/S008154381002015X.
9. Magaril-Il'yaev, G. G. and Sivkova, E. O. Optimal Recovery of the Semi-Group Operators from Inaccurate Data, Eurasian Mathematical Journal, 2019, vol. 10, no. 4, pp. 75-84. DOI: 10.32523/2077-9879-2019-10-4-75-84.
10. Sivkova, E. O. Optimal Recovery of a Family of Operators
from Inaccurate Measurements on a Compact, Vladikavkaz Mathematical Journal, 2023, vol. 25, no. 2, pp. 124-135 (in Russian). DOI: 10.46698/b9762-8415-3252-n.
11. Magaril-Il'yaev, G. G., Osipenko, K. Yu. and Tikhomirov, V. M.
On Optimal Recovery of Heat Equation Solutions, Approximation Theory: A Volume Dedicated to B. Bojanov, Eds. D. K. Dimitrov, G. Nikolov and R. Uluchev, Sofia, Marin Drinov Academic Publishing House, 2004, pp. 163-175.
12. Osipenko, K. Yu. On the Reconstruction of the Solution of the Dirichlet Problem from Inexact Initial Data, Vladikavkaz Mathematical Journal, 2004, vol. 6, no. 4, pp. 55-62 (in Russian).
13. Balova, E. A. Optimal Reconstruction of the Solution of the Dirichlet Problem From Inaccurate Input Data, Mathematical Notes, 2007, vol. 82, no. 3, pp. 285-294. DOI: 10.1134/S0001434607090015.
14. Magaril-Il'yaev, G. G. and Osipenko, K. Yu. Optimal Recovery of the Solution of the Heat Equation from Inaccurate Data, Sbornik: Mathematics, 2009, vol. 200, no. 5, pp. 665-682. DOI: 10.1070/SM2009v200n05ABEH004014.
15. Abramova, E. V. The Best Recovery of the Solution of the Dirichlet
Problem from Inaccurate Spectrum of the Boundary Function,
Vladikavkaz Mathematical Journal, 2017, vol. 19, no. 4, pp. 3-12 (in Russian).
DOI: 10.23671/VNC.2018.4.9163.
16. Balova, E. A. and Osipenko, K. Yu. Optimal Recovery Methods for Solutions of the Dirichlet Problem that are Exact on Subspaces of Spherical Harmonics, Mathematical Notes, 2018, vol. 104, no. 6, pp. 781-788.
DOI: 10.1134/S0001434618110238.
17. Abramova, E. V., Magaril-Il'yaev, G. G. and Sivkova, E. O.
Best Recovery of the Solution of the Dirichlet Problem
in a Half-Space from Inaccurate Data, Computational Mathematics and Mathematical Physics, 2020, vol. 60, no. 10, pp. 1656-1665. DOI: 10.1134/S0965542520100036.
