New methods for localizing discontinuities of a noisy function

For a problem of localizing singularities (discontinuities of the first kind) of a noisy function in L p (1 ≤ p < ∞), new classes of regularizing methods are constructed. The methods determine the number of discontinuities and approximate their positions. The upper and lower bounds of the localizing singularities and the separability threshold are also obtained. It is proved that the methods are order-optimal by accuracy and separability on some classes of functions with discontinuities.