We solve the extremal problem $$$| x^{(k)}_{pm} |_{L_p[a,b]} rightarrow sup$$$, $$$k = 0, 1, ..., r-1$$$, over the set of pairs $$$(x, I)$$$ of functions $$$xin W^r_{infty} (mathbb{R})$$$ and intervals $$$I = [a,b]$$$ with restrictions on the local norm of function $$$x$$$ and the measure of support $$$mu { mathrm{supp}_{[a,b]} x^{(k)}_{pm} }$$$.
{"title":"Bojanov-Naidenov problem for positive (negative) parts of differentiable functions on the real domain","authors":"V. V. Kameneva, V. Kofanov","doi":"10.15421/241804","DOIUrl":"https://doi.org/10.15421/241804","url":null,"abstract":"We solve the extremal problem $$$| x^{(k)}_{pm} |_{L_p[a,b]} rightarrow sup$$$, $$$k = 0, 1, ..., r-1$$$, over the set of pairs $$$(x, I)$$$ of functions $$$xin W^r_{infty} (mathbb{R})$$$ and intervals $$$I = [a,b]$$$ with restrictions on the local norm of function $$$x$$$ and the measure of support $$$mu { mathrm{supp}_{[a,b]} x^{(k)}_{pm} }$$$.","PeriodicalId":339757,"journal":{"name":"Dnipro University Mathematics Bulletin","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132807375","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the problem of approximation order of twice continuously differentiable functions of many variables by piecewise constants. We show that the saturation order of piecewise constant approximation in $$$L_p$$$ norm on convex partitions with $$$N$$$ cells is $$$N^{-2/(d+1)}$$$, where $$$d$$$ is the number of variables.
{"title":"Lower estimates on the saturation order of approximation of twice continuously differentiable functions by piecewise constants on convex partitions","authors":"O. Kozynenko","doi":"10.15421/241805","DOIUrl":"https://doi.org/10.15421/241805","url":null,"abstract":"We consider the problem of approximation order of twice continuously differentiable functions of many variables by piecewise constants. We show that the saturation order of piecewise constant approximation in $$$L_p$$$ norm on convex partitions with $$$N$$$ cells is $$$N^{-2/(d+1)}$$$, where $$$d$$$ is the number of variables.","PeriodicalId":339757,"journal":{"name":"Dnipro University Mathematics Bulletin","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127039195","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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