Geodesic
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, Tome 14 (2015), pp. 82-99

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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. 
@article{IIGUM_2015_14_a6,
     author = {N. A. Sidorov and D. N. Sidorov and I. R. Muftahov},
     title = {Perturbation theory and the {Banach--Steinhaus} theorem for regularization of the linear equations of the first kind},
     journal = {The Bulletin of Irkutsk State University. Series Mathematics},
     pages = {82--99},
     publisher = {mathdoc},
     volume = {14},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IIGUM_2015_14_a6/}
}
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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, Tome 14 (2015), pp. 82-99. http://geodesic.mathdoc.fr/item/IIGUM_2015_14_a6/