Regularizing algorithms for localizing the breakpoints of a noisy function

We consider the problem of localizing the singularities (breakpoints) of functions that are noisy in the spaces $L_p$, $1<p<\infty$, or $C$. We construct a wide class of smoothing algorithms that determine the number and location of breakpoints. In addition, for the case when a function is noisy in $C$, a finitedifference method is constructed. For the proposed methods, convergence theorems are proved and approximation accuracy estimates for the location of breakpoints are obtained. The lower estimates obtained in this paper show the order-optimality of the methods. For all the methods constructed, their capacity of separating close breakpoints is investigated.