{"title":"与最佳多项式近似有关的新估计","authors":"J. Bustamante","doi":"10.33993/jnaat521-1313","DOIUrl":null,"url":null,"abstract":"In some old results, we find estimates the best approximation \\(E_{n,p}(f)\\) of a periodic function satisfying \\(f^{(r)}\\in\\mathbb{L}^p_{2\\pi}\\) in terms of the norm of \\(f^{(r)}\\) (Favard inequality). In this work, we look for a similar result under the weaker assumption \\(f^{(r)}\\in \\mathbb{L}^q_{2\\pi}\\), with \\(1<q<p<\\infty\\). We will present inequalities of the form \\(E_{n,p}(f)\\leq C(n)\\Vert D^{(r)}f\\Vert_q\\), where \\(D^{(r)}\\) is a differential operator. We also study the same problem in spaces of non-periodic functions with a Jacobi weight.","PeriodicalId":287022,"journal":{"name":"Journal of Numerical Analysis and Approximation Theory","volume":"1 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2023-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"New estimates related with the best polynomial approximation\",\"authors\":\"J. Bustamante\",\"doi\":\"10.33993/jnaat521-1313\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In some old results, we find estimates the best approximation \\\\(E_{n,p}(f)\\\\) of a periodic function satisfying \\\\(f^{(r)}\\\\in\\\\mathbb{L}^p_{2\\\\pi}\\\\) in terms of the norm of \\\\(f^{(r)}\\\\) (Favard inequality). In this work, we look for a similar result under the weaker assumption \\\\(f^{(r)}\\\\in \\\\mathbb{L}^q_{2\\\\pi}\\\\), with \\\\(1<q<p<\\\\infty\\\\). We will present inequalities of the form \\\\(E_{n,p}(f)\\\\leq C(n)\\\\Vert D^{(r)}f\\\\Vert_q\\\\), where \\\\(D^{(r)}\\\\) is a differential operator. We also study the same problem in spaces of non-periodic functions with a Jacobi weight.\",\"PeriodicalId\":287022,\"journal\":{\"name\":\"Journal of Numerical Analysis and Approximation Theory\",\"volume\":\"1 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-07-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Numerical Analysis and Approximation Theory\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.33993/jnaat521-1313\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Numerical Analysis and Approximation Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.33993/jnaat521-1313","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
New estimates related with the best polynomial approximation
In some old results, we find estimates the best approximation \(E_{n,p}(f)\) of a periodic function satisfying \(f^{(r)}\in\mathbb{L}^p_{2\pi}\) in terms of the norm of \(f^{(r)}\) (Favard inequality). In this work, we look for a similar result under the weaker assumption \(f^{(r)}\in \mathbb{L}^q_{2\pi}\), with \(1
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