Solving equations of the first kind on classes of functions with singularities. (English) Zbl 1125.65320

Antonova, T. V.

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].

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)

Keywords:

convolution equations; ill-posed problems