Investigation of Methods of Localization of q-Jumps and Discontinuities of First Kind of Noisy Function

We consider the problem of localizing (determination of position) the first kind discontinuities of a function of one variable and the problem of localizing q-jumps of a noisy function. In the first case, we assume that the exact function is smooth except for a finite number of discontinuities of the first kind. In the second case, the exact function is smooth except for a finite number of small segments of length 2q. It is required to determine the number of discontinuities (q-jumps) and approximate their positions using the function approximately specified in L2(ℝ) and the level of disturbance. We construct a class of regular averaging methods and obtain estimates of the accuracy of localization, separability, and observability on classes of correctness.