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].

For the entire collection see [Zbl 1116.90003].

##### MSC:

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) |