Abstract
The aim of this note is to discuss the convergence properties of various methods for the approximate solution of ill-posed equations. If an operator T between Banach spaces has a non-closed range, then there exists no linear uniformly convergent approximation, method, but at most pointwise approximation methods. for the approximate solution of an equation (1) Tx = y. These methods in general converge arbitrarily slow, i.e. there may exist some y for which the approximations have a good convergence rate, but in general for each order of convergence there are right hand sides y with a worse convergence rate. This phenomenon is not restricted to ill-posed equations. Recently I have shown [121 that many common approximation schemes for integral equations of the second kind show arbitrarily slow convergence, too.
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© 1985 Birkhäuser Verlag Basel
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Schock, E. (1985). Approximate Solution of Ill-Posed Equations: Arbitrarily Slow Convergence vs. Superconvergence. In: Hämmerlin, G., Hoffmann, KH. (eds) Constructive Methods for the Practical Treatment of Integral Equations. International Series of Numerical Mathematics, vol 73. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9317-6_20
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DOI: https://doi.org/10.1007/978-3-0348-9317-6_20
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