{"title":"Approximation to the Derivatives of a Function Defined on a Simplex under Lagrangian Interpolation","authors":"N. V. Baidakova, Yu. N. Subbotin","doi":"10.1134/s0001434624010012","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> New upper bounds are found in the problem of approximation to the <span>\\(k\\)</span>th derivatives of a function of <span>\\(d\\)</span> variables defined on a simplex by the derivatives of an algebraic polynomial of degree at most <span>\\(n\\)</span> (<span>\\(0\\le k\\le n\\)</span>) interpolating the values of the function at equidistant nodes of the simplex. The estimates are obtained in terms of the diameter of the simplex, the angular characteristic introduced in the paper, the dimension <span>\\(d\\)</span>, the degree <span>\\(n\\)</span> of the polynomial, and the order <span>\\(k\\)</span> of the derivative to be estimated and do not contain unknown parameters. These estimates are compared with those most frequently used in the literature. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":"32 1","pages":""},"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/s0001434624010012","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
New upper bounds are found in the problem of approximation to the \(k\)th derivatives of a function of \(d\) variables defined on a simplex by the derivatives of an algebraic polynomial of degree at most \(n\) (\(0\le k\le n\)) interpolating the values of the function at equidistant nodes of the simplex. The estimates are obtained in terms of the diameter of the simplex, the angular characteristic introduced in the paper, the dimension \(d\), the degree \(n\) of the polynomial, and the order \(k\) of the derivative to be estimated and do not contain unknown parameters. These estimates are compared with those most frequently used in the literature.
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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