{"title":"On Local Stability in the Complete Prony Problem","authors":"A. A. Lomov","doi":"10.1134/s1055134424020044","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We consider the variational Prony problem on approximating observations\n<span>\\(x \\)</span> by the sum of exponentials. We find critical points\nand the second derivatives of the implicit function <span>\\(\\theta \\)</span> that relates perturbation in <span>\\(x \\)</span> with the corresponding exponents. We suggest\nupper bounds for the second order increments and describe the domain, where the accuracy of\na linear approximation of <span>\\(\\theta \\)</span> is acceptable.\nWe deduce lower estimates of the norm of deviation of <span>\\(\\theta \\)</span> for small perturbations in <span>\\(x \\)</span>. We compare our estimates of this norm with\nupper bounds obtained with the use of Wilkinson’s inequality.\n</p>","PeriodicalId":39997,"journal":{"name":"Siberian Advances in Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Siberian Advances in Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1134/s1055134424020044","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We consider the variational Prony problem on approximating observations
\(x \) by the sum of exponentials. We find critical points
and the second derivatives of the implicit function \(\theta \) that relates perturbation in \(x \) with the corresponding exponents. We suggest
upper bounds for the second order increments and describe the domain, where the accuracy of
a linear approximation of \(\theta \) is acceptable.
We deduce lower estimates of the norm of deviation of \(\theta \) for small perturbations in \(x \). We compare our estimates of this norm with
upper bounds obtained with the use of Wilkinson’s inequality.
期刊介绍：
Siberian Advances in Mathematics is a journal that publishes articles on fundamental and applied mathematics. It covers a broad spectrum of subjects: algebra and logic, real and complex analysis, functional analysis, differential equations, mathematical physics, geometry and topology, probability and mathematical statistics, mathematical cybernetics, mathematical economics, mathematical problems of geophysics and tomography, numerical methods, and optimization theory.
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