{"title":"Yu. N. Subbotin’s Method in the Problem of Extremal Interpolation in the Mean in the Space $$L_p(\\mathbb R)$$ with Overlapping Averaging Intervals","authors":"V. T. Shevaldin","doi":"10.1134/s0001434624050365","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> On a uniform grid on the real axis, we study the Yanenko–Stechkin–Subbotin problem of extremal function interpolation in the mean in the space <span>\\(L_p(\\mathbb R)\\)</span>, <span>\\(1<p<\\infty\\)</span>, of two-way real sequences with the least value of the norm of a linear formally self-adjoint differential operator <span>\\({\\mathcal L}_n\\)</span> of order <span>\\(n\\)</span> with constant real coefficients. In case of even <span>\\(n\\)</span>, the value of the least norm in the space <span>\\(L_p(\\mathbb R)\\)</span>, <span>\\(1<p<\\infty\\)</span>, of the extremal interpolant is calculated exactly if the grid step <span>\\(h\\)</span> and the averaging step <span>\\(h_1\\)</span> are related by the inequality <span>\\(h<h_1\\le 2h\\)</span>. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Notes","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s0001434624050365","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
On a uniform grid on the real axis, we study the Yanenko–Stechkin–Subbotin problem of extremal function interpolation in the mean in the space \(L_p(\mathbb R)\), \(1<p<\infty\), of two-way real sequences with the least value of the norm of a linear formally self-adjoint differential operator \({\mathcal L}_n\) of order \(n\) with constant real coefficients. In case of even \(n\), the value of the least norm in the space \(L_p(\mathbb R)\), \(1<p<\infty\), of the extremal interpolant is calculated exactly if the grid step \(h\) and the averaging step \(h_1\) are related by the inequality \(h<h_1\le 2h\).
期刊介绍：
Mathematical Notes is a journal that publishes research papers and review articles in modern algebra, geometry and number theory, functional analysis, logic, set and measure theory, topology, probability and stochastics, differential and noncommutative geometry, operator and group theory, asymptotic and approximation methods, mathematical finance, linear and nonlinear equations, ergodic and spectral theory, operator algebras, and other related theoretical fields. It also presents rigorous results in mathematical physics.
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