{"title":"A Generalized Translation Operator Generated by the Sinc Function on an Interval","authors":"V. V. Arestov, M. V. Deikalova","doi":"10.1134/s0081543823060032","DOIUrl":null,"url":null,"abstract":"<p>We discuss the properties of the generalized translation operator generated by the system of functions <span>\\(\\mathfrak{S}=\\{{(\\sin k\\pi x)}/{(k\\pi x)}\\}_{k=1}^{\\infty}\\)</span> in the spaces <span>\\(L^{q}=L^{q}((0,1),{\\upsilon})\\)</span>, <span>\\(q\\geq 1\\)</span>, on the interval <span>\\((0,1)\\)</span> with the weight <span>\\(\\upsilon(x)=x^{2}\\)</span>. We find an integral representation of this operator and study its norm in the spaces <span>\\(L^{q}\\)</span>, <span>\\(1\\leq q\\leq\\infty\\)</span>. The translation operator is applied to the study of Nikol’skii’s inequality between the uniform norm and the <span>\\(L^{q}\\)</span>-norm of polynomials in the system <span>\\(\\mathfrak{S}\\)</span>.\n</p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"64 1","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the Steklov Institute of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s0081543823060032","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We discuss the properties of the generalized translation operator generated by the system of functions \(\mathfrak{S}=\{{(\sin k\pi x)}/{(k\pi x)}\}_{k=1}^{\infty}\) in the spaces \(L^{q}=L^{q}((0,1),{\upsilon})\), \(q\geq 1\), on the interval \((0,1)\) with the weight \(\upsilon(x)=x^{2}\). We find an integral representation of this operator and study its norm in the spaces \(L^{q}\), \(1\leq q\leq\infty\). The translation operator is applied to the study of Nikol’skii’s inequality between the uniform norm and the \(L^{q}\)-norm of polynomials in the system \(\mathfrak{S}\).
期刊介绍:
Proceedings of the Steklov Institute of Mathematics is a cover-to-cover translation of the Trudy Matematicheskogo Instituta imeni V.A. Steklova of the Russian Academy of Sciences. Each issue ordinarily contains either one book-length article or a collection of articles pertaining to the same topic.