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Moment inversion problem for piecewise \(D\)-finite functions. (English) Zbl 1193.94011
The problem of reconstruction of univariate functions with jump discontinuities (discontinuities of the first kind) at unknown positions from their moments is studied. Such functions are assumed to satisfy an a priori unknown linear homogeneous differential equation with polynomial coefficients on each continuity interval. Such univariate functions can be specified by a finite amount of information. The principal result of this paper is a generic algorithm for reconstructing a piecewise \(D\)-finite (“differentiably” finite, see R. P. Stanley [Eur. J. Comb. 1, 175–188 (1980; Zbl 0445.05012)]) function from its moments which leads to the design of inversion algorithm in noise environment. The performance tests have been fulfilled on synthetic data presenting the several types of signals: piecewise sinusoid, piecewise-polynomial function and a rational function without discontinuities applied. The theoretical stability estimate for the single special case of linear combinations of Dirac \(\delta\)-functions is presented. It is to be noted that practically, these studies are related to development of the theory of regularizing algorithms for localization of breakpoints of noisy functions in \(L_2\) which have been recently addressed by series of publications of A. L. Ageev and T. V. Antonova [see e.g. J. Inverse Ill-Posed Probl. 16, No. 7, 639–650 (2008: Zbl 1158.65087)].
As a footnote, it is worth mentioning that singularities localization is very important in practical applications in DSP, such as computer tomography and specifically in medical imaging and filters design, where jump discontinuities (edges) represent crucial information of the signal.

94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
15A29 Inverse problems in linear algebra
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
47A52 Linear operators and ill-posed problems, regularization
65F22 Ill-posedness and regularization problems in numerical linear algebra
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