Regularizing algorithms for localization of breakpoints of noisy functions

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

“…The third variant with exactly l discontinuities was considered earlier in [12][13][14][15][16][17]. In this variant, condition (2) holds for γ = 0 automatically; therefore, only condition (1) was used in these papers.…”

“…Note that condition (1) was used in [12][13][14][15][16][17]. As for the condition of "smallness" of the remaining discontinuities, it is clear that condition (2) is too complicated and nontransparent.…”

“…In contrast to [12][13][14][15][16][17], in the present paper, we substantially use facts from the theory of the Lebesgue-Stieltjes integral [18], which allows us to justify the main decomposition for an auxiliary function by references to known results. Note also that [17] is the key paper that enabled us to consider the new statement.…”

“…In the meantime, in some applications, for example, in signal processing, it is natural to assume that the function x does not decrease at ±∞, i.e., does not belong to L p (−∞, +∞) and has countably many discontinuities of the first kind. Therefore, it is reasonable to investigate other classes of functions with countably many singularities and carry over the methods of [12][13][14][15][16][17] to this case. As in [12][13][14][15][16][17], we assume that, instead of the exact function x, the error level δ and an approximate function x δ are known…”

“…The other large cycle of papers [12][13][14][15][16][17] (see also [4]) on the localization of singularities was implemented for problems in the deterministic formulation. In these papers, it is assumed that an exact function x belongs to L 2 (−∞, +∞) (or L p (−∞, +∞), 1 < p < ∞), has a finite number of discontinuities of the first kind, and is smooth outside the discontinuities.…”