Geodesic
Estimation of the spectrum of discrete sequences in ill-posed problems based on the study of the numerical rank of the trajectory matrix
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations and Optimal Control, Tome 183 (2020), pp. 73-84

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In this paper, we discuss properties of the singular value decomposition (SVD-decomposition) within the framework of the analysis of the numerical rank in ill-posed problems for determining frequency properties of discrete sequences consisting of trigonometric binomials. The properties of the numerical rank of the SVD-decomposition are indicated. We propose an algorithm for determining the frequencies of trigonometric binomials involved in the original function that forms a discrete sequence; this algorithm is based on estimators of the numerical rank. We obtain a stable criterion for estimating the numerical rank based on the arithmetic-mean estimators of the pseudo-null-space of the trajectory matrix. Also, we present the results of numerical experiments that demonstrate the consistency of the arithmetic-mean estimator of the pseudo-zero-space for the analysis of the spectrum of noisy sequences.
Mots-clés : singular value decomposition
Keywords: spectral analysis, numerical rank, ill-posed problem, analysis of frequency components. 
@article{INTO_2020_183_a6,
     author = {V. S. Kedrin},
     title = {Estimation of the spectrum of discrete sequences in ill-posed problems based on the study of the numerical rank of the trajectory matrix},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {73--84},
     publisher = {mathdoc},
     volume = {183},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2020_183_a6/}
}
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V. S. Kedrin. Estimation of the spectrum of discrete sequences in ill-posed problems based on the study of the numerical rank of the trajectory matrix. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations and Optimal Control, Tome 183 (2020), pp. 73-84. http://geodesic.mathdoc.fr/item/INTO_2020_183_a6/