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Solving equations of the first kind on classes of functions with singularities. (English) Zbl 1125.65320
Eremin, I. I. (ed.), Mathematical programming, regularization and approximation. Transl. from the Russian. Moscow: Maik Nauka/Interperiodica. Proceedings of the Steklov Institute of Mathematics 2002, Suppl. 1, S145-S189 (2002).
Summary: Linear and nonlinear ill-posed problems are considered on classes of functions with a finite number of singularities (discontinuities of the first kind or \(\delta\)-functions). A special technique is elaborated for investigating phenomena that arise in a neighborhood of singularities of a function and are similar to the Gibbs phenomenon. This technique allows one to construct approximations of the characteristics of singularities and to approximate the required function outside a small neighborhood of singularity points in the uniform metric. The following unstable problems are considered for these classes of functions: recovering a function by noised data in \(L_2\); solving linear equations of the first kind of convolution type and equations of the first kind of convolution type with a kernel depending on an unknown parameter. For all the values being defined, estimates of accuracy are given.
For the entire collection see [Zbl 1116.90003].

65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
47A52 Linear operators and ill-posed problems, regularization
47J06 Nonlinear ill-posed problems
65R30 Numerical methods for ill-posed problems for integral equations
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)