Localization of boundaries for subsets of discontinuity points of noisy function

A. L. Ageev^ab, T. V. Antonova<sup>a

a</sup> Institute of Mathematics and Mechanics, Ural Branch of Russian Academy of Sciences, 16 S. Kovalevskaya str., Ekaterinburg, 620990 Russia
^b Ural Federal University, 19 Mira str., Ekaterinburg, 620002 Russia

Abstract: We consider ill-posed problem of localization of discontinuities of the first kind of function in one variable under condition that in metric $L_2$ there are given approximately measured function and a level of inaccuracy. We propose a new statement of the problem when all discontinuities can be divided into subsets and localization is performed for subsets of discontinuities. Under additional assumption that all discontinuities possess jumps of one sign we construct new regular method which admits to determine a number of subsets of discontinuities and to approximate their boundaries with estimation of approximation accuracy.

Keywords: ill-posed problem, regularizing algorithm, discontinuity of the first kind, subsets of discontinuity points, localization of boundaries of subsets of singularities, separability threshold.

UDC: 517.988

Received: 01.07.2016

 English version: Russian Mathematics (Izvestiya VUZ. Matematika), 2017, 61:11, 10–15

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