New methods for localizing discontinuities of a noisy function

For a problem of localizing the singularities (discontinuities of the first kind) of a noisy function in $L_p$ ($1\le p<\infty$), new classes of regularizing methods are constructed. These methods determine the number of discontinuities and approximate their positions. Also, upper and lower bound of the localizing singularities and the separability threshold, is obtain. It is proved that the methods are order-optimal by accuracy as well as separability on some classes of functions with discontinuities.