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 The Bulletin of Irkutsk State University. Series Mathematics, 2015, Volume 14, Pages 82–99 (Mi iigum245)

Perturbation theory and the Banach–Steinhaus theorem for regularization of the linear equations of the first kind

N. A. Sidorova, D. N. Sidorovab, I. R. Muftahovc

a Irkutsk State University, 1, K. Marx st., Irkutsk, 664003
b Melentiev Energy Systems Institute of Siberian Branch of the Russian Academy of Science (ESI SB RAS), 130, Lermontov st., Irkutsk, 664033
c National Research Irkutsk State Technical University, 80, Lermontov st., Irkutsk, 664033

Abstract: The regularizing equations with a vector parameter of regularization are constructed for the linear equations with closed operator acting in Banach spaces. Range of the operator can be an open, and the homogeneous equation may have a non-trivial solution. It is assumed that only approximations of operator and source are known. The conditions of solution uniqueness for the auxiliary regularized equation are derived. The convergence of regularized solution to B-normal solution of the exact equation is proved. The bounds estimates are derived for both deterministic and stochastic cases. The choice of the stabilizing operator and vector regularization parameter are provided. The method is applied to the problem of stable differentiation.

Keywords: Regularizing Equation, $\delta$-approximation, Banach–Steinhaus Theorem, Perturbation Theory, Inverse Problems, Regularization, Expectation, Perturbation Theory, Stable Differentiation.

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Citation: N. A. Sidorov, D. N. Sidorov, I. R. Muftahov, “Perturbation theory and the Banach–Steinhaus theorem for regularization of the linear equations of the first kind”, The Bulletin of Irkutsk State University. Series Mathematics, 14 (2015), 82–99

Citation in format AMSBIB
\Bibitem{SidSidMuf15} \by N.~A.~Sidorov, D.~N.~Sidorov, I.~R.~Muftahov \paper Perturbation theory and the Banach--Steinhaus theorem for regularization of the linear equations of the first kind \jour The Bulletin of Irkutsk State University. Series Mathematics \yr 2015 \vol 14 \pages 82--99 \mathnet{http://mi.mathnet.ru/iigum245} 

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This publication is cited in the following articles:
1. I. R. Muftakhov, D. N. Sidorov, N. A. Sidorov, “O regulyarizatsii po Lavrentevu integralnykh uravnenii pervogo roda v prostranstve nepreryvnykh funktsii”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 15 (2016), 62–77
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