On the localization of singularities of the first kind for a function of bounded variation

Methods of the localization (detection) of discontinuities of the first kind for a function of bounded variation of one variable are constructed and investigated. We consider the problem of localizing discontinuities of a function that is noisy in the space $L_2(-\infty,+\infty)$. We distinguish between discontinuities with the absolute value of the jump greater than some positive $\Delta^{\min}$ and discontinuities satisfying a smallness condition for the value of the jump. It is required to find the number of discontinuities and localize them using the approximately given function and the error level. Since the problem is ill-posed, regularizing algorithms should be used for its solution. Under additional conditions on the exact function, we construct regular methods for the localization of discontinuities and obtain estimates for the accuracy of localization and for the separability threshold, which is another important characteristic of the method. The (order) optimality of the constructed methods on the classes of functions with singularities is established.