On the localization of discontinuities of the first kind for a function of bounded variation

Abstract

Methods of the localization (detection of positions) of discontinuities of the first kind for a univariate function of bounded variation are constructed and investigated. Instead of an exact function, its approximation in L2(-∞,+∞) and the error level are known. We divide the discontinuities into two sets, one of which contains discontinuities with the absolute value of the jump greater than some positive Δmin; the other set contains discontinuities satisfying a smallness condition for the value of the jump. It is required to find the number of discontinuities in the former set and localize them using the approximately given function and the error level. Since the problem is ill-posed, regularizing algorithms should be used for its solution. Under additional conditions on the exact function, we construct regular methods for the localization of discontinuities and obtain estimates for the accuracy of localization and for the separability threshold, which is another important characteristic of the method. The (order) optimality of the constructed methods on the classes of functions with singularities is established. © 2013 Pleiades Publishing, Ltd.