### 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.

### Full Text

### 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},
}

### Google Scholar Metrics

Citing articles in Google Scholar

Similar articles in Google Scholar

### Export article

Save to Mendeley