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Non-strongly converging approximation methods. (English) Zbl 0696.65055

To solve operator equations of the form (*) \(Ax=y\) one may consider the problem \((+)\) \(P_{\tau}Ax_{\tau}=P_{\tau}y\), where \(P_{\tau}\) are projection operators. If there exists a \(\tau_ 0\) such that for \(\tau \geq \tau_ 0\) the problem \((+)\) has a unique solution \(x_{\tau}\), the convergence \(x_{\tau}\to x\) must be guaranteed. There are interesting examples, e.g. in the case of integral equations on the space of bounded measurable functions where the convergence only holds in a weaker sense. This observation leads the authors to the definition of the convergence with respect to a family of projections. Convergent approximation methods for (*) in this weaker sense are studied. The result is applied to the Wiener-Hopf integral equation using a composite quadrature rule method.
Reviewer: W.Petry

MSC:

65J10 Numerical solutions to equations with linear operators
65R20 Numerical methods for integral equations
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
47A50 Equations and inequalities involving linear operators, with vector unknowns
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