Approximation of function with finite number of discontinuities by noised data

Abstract

In this paper we consider the ill-posed problem of recovery of a function which has a finite number of jumps using noised data. The method which we suggest allows us to find a number of discontinuities. The position of the discontinuity points and the values of jumps are derived approximately. The estimates of accuracy of positions and jumps values are given. Approximation of initial function is constructed. The uniform estimate of proximity of the approximate function and the exact function is given outside the neighborhoods discontinuity points.

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

“…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.…”

“…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.…”

“…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.…”

“…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…”

“…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.…”