Estimates of Characteristics of Localization Methods for Discontinuities of the First Kind of a Noisy Function

Abstract

This is theoretical study of the ill-posed problem on localization (determination of position) of discontinuities of the first kind of a function of one variable. The exact function x is smooth but has finitely many discontinuities of the first kind. Given some approximate function xδ, ||xδ − x|| L2(ℝ) ≤ δ, and the error level δ, it is required to determine the number of discontinuities and approximate their location with an estimate of the approximation accuracy. Regular localization methods are constructed on the basis of averages that are scaled by the regularization parameter. The investigation of these methods consists in carrying out estimates for their three main characteristics on the classes of correctness: accuracy of localization, separability, and observability. Under consideration is the general formulation of the problem that generalizes the previously obtained results. The necessary conditions are obtained that must be satisfied by the accuracy of localization, separability, and observability. Also, the sufficient conditions close to the necessary are found, under which a localization method is constructed with the specified accuracy, observability, and separability. The concept of optimality of the localization methods is introduced in terms of the order of accuracy, separability, and observability (in the whole) and the methods are constructed that are optimal in order in the whole.