On ill-posed problems of localization of singularities

Ill-posed problems of approximating (localizing) the positions of isolated singularities of a function of one variable are discussed. The function either is given with an error or is a solution to the convolution-type Fredholm integral equation of the first kind with an error in the right-hand side. The singularities can be $\delta$-functions, discontinuities of the first kind, or breakpoints. Earlier, the authors proposed an approach to deriving accuracy estimates for localization algorithms, which is similar to the classical approach of investigating methods on correctness classes. As a development of this theory, a general scheme of construction and investigation is proposed for regular method of localizing the singularities. The scheme can be used to uniformly derive many of the known results as well as new statements. Several classes of regularization methods generated by averaging kernels are considered. Estimates of localization accuracy and estimates of another important characteristic of the methods, namely, of the separability threshold, are obtained for the proposed methods. Lower estimates for the attainable accuracy and separability are obtained, which allows to establish the (order) optimality of the constructed methods on classes of functions with singularities for some problems.