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# On the approximation to solutions of operator equations by the least squares method

### Abstract

We consider the equation $Au = f$, where $A$ is a linear operator with compact inverse in a Hilbert space. For the approximate solution $u_n$ of this equation by the least squares method in a coordinate system that is an orthonormal basis of eigenvectors of a self-adjoint operator $B$ similar to $A \ ({\mathcal{D}} (A) = {\mathcal{D}} (B))$, we give a priori estimates for the asymptotic behavior of the expression $R_n = \|Au_n - f\|$ as $n \to \infty$. A relationship between the order of smallness of this expression and the degree of smoothness of the solution $u$ with respect to the operator $B$ (direct and converse theorems) is established.

### Article Information

 Title On the approximation to solutions of operator equations by the least squares method Source Methods Funct. Anal. Topology, Vol. 14 (2008), no. 3, 229-241 MathSciNet MR2458488 Copyright The Author(s) 2008 (CC BY-SA)

### Authors Information

Myroslav L. Gorbachuk
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, Ukraine

Valentyna I. Gorbachuk
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, Ukraine

### Citation Example

Myroslav L. Gorbachuk and Valentyna I. Gorbachuk, On the approximation to solutions of operator equations by the least squares method, Methods Funct. Anal. Topology 14 (2008), no. 3, 229-241.

### BibTex

@article {MFAT480,
AUTHOR = {Gorbachuk, Myroslav L. and Gorbachuk, Valentyna I.},
TITLE = {On the approximation to solutions of operator equations by the least squares method},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {14},
YEAR = {2008},
NUMBER = {3},
PAGES = {229-241},
ISSN = {1029-3531},
URL = {http://mfat.imath.kiev.ua/article/?id=480},
}