Geodesic
On the Approximation to Solutions of Operator Equations by the Least Squares Method
Funkcionalʹnyj analiz i ego priloženiâ, Tome 39 (2005) no. 1, pp. 85-90

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We consider the equation $Au=f$, where $A$ is a linear operator with compact inverse $A^{-1}$ in a separable Hilbert space $\mathfrak{H}$. For the approximate solution $u_n$ of this equation by the least squares method in a coordinate system $\{e_k\}_{k\in\mathbb{N}}$ that is an orthonormal basis of eigenvectors of a self-adjoint operator $B$ similar to $A$ ($\mathcal{D}(B)=\mathcal{D}(A)$), we give a priori estimates for the asymptotic behavior of the expressions $r_n=\|u_n-u\|$ and $R_n=\|Au_n-f\|$ as $n\to\infty$. A relationship between the order of smallness of these expressions and the degree of smoothness of $u$ with respect to the operator $B$ is established.
Keywords: Hilbert space, operator equation, similar operator, approximate solution, least squares method, coordinate system, a priori estimate, closed operator, smooth vector, analytic vector, entire vector, entire vector of exponential type. 
M. L. Gorbachuk. On the Approximation to Solutions of Operator Equations by the Least Squares Method. Funkcionalʹnyj analiz i ego priloženiâ, Tome 39 (2005) no. 1, pp. 85-90. http://geodesic.mathdoc.fr/item/FAA_2005_39_1_a7/
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