The abstract theory of approximate methods for solving operator equations
The author presents a certain extension of the abstract theory of approximate schemes formulated in the book of \textit{V. A. Trenogin} [Functional analysis (Nauka 1980; Zbl 0517.46001)]. The exact and the approximate equations are considered in different spaces, \(A\in L(X,Y)\) and \(\overline A\in L(\overline X,\overline Y)\). Therefore, corresponding approximation and interpolation operators \(T_X:X\to\overline X\) and \(S_X:\overline X\to X\) for the spaces \(X\), \(\overline X\) and \(T_Y\), \(S_Y\) for the spaces \(Y, \overline Y\) are applied. Together with T-convergence (corresponding to the difference \(\|T_Xx-\overline x\|_{\overline X}\)) considered in the mentioned book, S-convergence (corresponding to \(\|x-S_X\overline x\|\)) is investigated. The conditions under which the invertibility of the operator \(A\) (or \(\overline A\)) implies the invertibility of the operator \(\overline A\) (or \(A\)) are formulated. Moreover, theorems concerning \(T\)-convergence as well as \(S\)-convergence of abstract approximate methods are obtained.
scientific article; zbMATH DE number 1541591