Investigation of methods of localization of $q$-jumps and discontinities of firsth king of noisy function

We consider the problem of localizing (determination of position) of discontinuities of the first kind of a function of one variable and the problem of localizing $q$-jumps of a noisy function. In the first case it is assumed 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 that the number of discontinuities ($q$-jumps) be determined and approximated their position from an approximately given function and the level of the perturbation in $L_2(\mathbb{R})$. We construct a class of regular averaging methods and obtain and estimates of the accuracy of localization, separability, and observability on classes of correctness.