{"title":"Extremal Interpolation in the Mean in the Space $$L_1(\\mathbb R)$$ with Overlapping Averaging Intervals","authors":"V. T. Shevaldin","doi":"10.1134/s0001434624010097","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> On a uniform grid on the real axis <span>\\(\\mathbb R\\)</span>, we study the Yanenko–Stechkin–Subbotin problem of extremal function interpolation in the mean in the space <span>\\(L_1(\\mathbb R)\\)</span> of two-way real sequences with the least value of the norm of a linear formally s This problem is considered for the class of sequences for which the generalized finite differences of order <span>\\(n\\)</span> corresponding to the operator <span>\\(\\mathcal L_n\\)</span> are bounded in the space <span>\\(l_1\\)</span>. In this paper, the least value of the norm is calculated exactly if the grid step <span>\\(h\\)</span> and the averaging step <span>\\(h_1\\)</span> of the function to be interpolated in the mean are related by the inequalities <span>\\(h<h_1\\le 2h\\)</span>. The paper is a continuation of the research by Yu. N. Subbotin and the author in this problem, initiated by Yu. N. Subbotin in 1965. The result obtained is new, in particular, for the <span>\\(n\\)</span>-times differentiation operator <span>\\(\\mathcal L_n(D)=D^n\\)</span>. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-04-22","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/s0001434624010097","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
On a uniform grid on the real axis \(\mathbb R\), we study the Yanenko–Stechkin–Subbotin problem of extremal function interpolation in the mean in the space \(L_1(\mathbb R)\) of two-way real sequences with the least value of the norm of a linear formally s This problem is considered for the class of sequences for which the generalized finite differences of order \(n\) corresponding to the operator \(\mathcal L_n\) are bounded in the space \(l_1\). In this paper, the least value of the norm is calculated exactly if the grid step \(h\) and the averaging step \(h_1\) of the function to be interpolated in the mean are related by the inequalities \(h<h_1\le 2h\). The paper is a continuation of the research by Yu. N. Subbotin and the author in this problem, initiated by Yu. N. Subbotin in 1965. The result obtained is new, in particular, for the \(n\)-times differentiation operator \(\mathcal L_n(D)=D^n\).
Abstract On a uniform grid on the real axis \(\mathbb R\)、我们研究了Yanenko-Stechkin-Subbotin问题，即在双向实数序列的均值空间\(L_1(\mathbb R)\)中的极值函数插值与线性形式的最小规范值。在本文中，如果要插值的函数在均值中的网格步长（h）和平均步长（h_1）通过不等式（h<h_1le 2h）相关，则可以精确计算出最小值的规范。本文是 Yu.N. Subbotin 和作者在这一问题上的研究的延续。N. Subbotin 于 1965 年开始的。所得到的结果是新的，特别是对于 \(n\)-times 微分算子 \(\mathcal L_n(D)=D^n\).
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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