{"title":"论加权大勒贝格空间中正弦和余弦函数的指数和三角函数系统的基础性质","authors":"M. I. Ismailov, I. F. Aliyarova","doi":"10.3103/s0027132224700128","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The paper is focused on the basis property of the system of exponentials and trigonometric systems of sine and cosine functions in a separable subspace of the weighted grand Lebesgue space generated by the shift operator. In this paper, with the help of the shift operator, a separable subspace <span>\\(G_{p),\\rho}(a,b)\\)</span> of the weighted space of the grand Lebesgue space <span>\\(L_{p),\\rho}(a,b)\\)</span> is defined. The density in <span>\\(G_{p),\\rho}(a,b)\\)</span> of the set <span>\\(G_{0}^{\\infty}([a,b])\\)</span> of infinitely differentiable functions that are finite on <span>\\([a,b]\\)</span> is studied. It is proved that if the weight function <span>\\(\\rho\\)</span> satisfies the Mackenhoupt condition, then the system of exponentials <span>\\(\\left\\{e^{int}\\right\\}_{n\\in Z}\\)</span> forms a basis in <span>\\(G_{p),\\rho}(-\\pi,\\pi)\\)</span>, and trigonometric systems of sine <span>\\(\\left\\{\\sin nt\\right\\}_{n\\geqslant 1}\\)</span> and cosine <span>\\(\\left\\{\\cos nt\\right\\}_{n\\geqslant 0}\\)</span> functions form bases in <span>\\(G_{p),\\rho}(0,\\pi)\\)</span>.</p>","PeriodicalId":42963,"journal":{"name":"Moscow University Mathematics Bulletin","volume":null,"pages":null},"PeriodicalIF":0.2000,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the Basis Property of the System of Exponentials and Trigonometric Systems of Sine and Cosine Functions in Weighted Grand Lebesgue Spaces\",\"authors\":\"M. I. Ismailov, I. F. Aliyarova\",\"doi\":\"10.3103/s0027132224700128\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<h3 data-test=\\\"abstract-sub-heading\\\">Abstract</h3><p>The paper is focused on the basis property of the system of exponentials and trigonometric systems of sine and cosine functions in a separable subspace of the weighted grand Lebesgue space generated by the shift operator. In this paper, with the help of the shift operator, a separable subspace <span>\\\\(G_{p),\\\\rho}(a,b)\\\\)</span> of the weighted space of the grand Lebesgue space <span>\\\\(L_{p),\\\\rho}(a,b)\\\\)</span> is defined. The density in <span>\\\\(G_{p),\\\\rho}(a,b)\\\\)</span> of the set <span>\\\\(G_{0}^{\\\\infty}([a,b])\\\\)</span> of infinitely differentiable functions that are finite on <span>\\\\([a,b]\\\\)</span> is studied. It is proved that if the weight function <span>\\\\(\\\\rho\\\\)</span> satisfies the Mackenhoupt condition, then the system of exponentials <span>\\\\(\\\\left\\\\{e^{int}\\\\right\\\\}_{n\\\\in Z}\\\\)</span> forms a basis in <span>\\\\(G_{p),\\\\rho}(-\\\\pi,\\\\pi)\\\\)</span>, and trigonometric systems of sine <span>\\\\(\\\\left\\\\{\\\\sin nt\\\\right\\\\}_{n\\\\geqslant 1}\\\\)</span> and cosine <span>\\\\(\\\\left\\\\{\\\\cos nt\\\\right\\\\}_{n\\\\geqslant 0}\\\\)</span> functions form bases in <span>\\\\(G_{p),\\\\rho}(0,\\\\pi)\\\\)</span>.</p>\",\"PeriodicalId\":42963,\"journal\":{\"name\":\"Moscow University Mathematics Bulletin\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.2000,\"publicationDate\":\"2024-07-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Moscow University Mathematics Bulletin\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.3103/s0027132224700128\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Moscow University Mathematics Bulletin","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3103/s0027132224700128","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
On the Basis Property of the System of Exponentials and Trigonometric Systems of Sine and Cosine Functions in Weighted Grand Lebesgue Spaces
Abstract
The paper is focused on the basis property of the system of exponentials and trigonometric systems of sine and cosine functions in a separable subspace of the weighted grand Lebesgue space generated by the shift operator. In this paper, with the help of the shift operator, a separable subspace \(G_{p),\rho}(a,b)\) of the weighted space of the grand Lebesgue space \(L_{p),\rho}(a,b)\) is defined. The density in \(G_{p),\rho}(a,b)\) of the set \(G_{0}^{\infty}([a,b])\) of infinitely differentiable functions that are finite on \([a,b]\) is studied. It is proved that if the weight function \(\rho\) satisfies the Mackenhoupt condition, then the system of exponentials \(\left\{e^{int}\right\}_{n\in Z}\) forms a basis in \(G_{p),\rho}(-\pi,\pi)\), and trigonometric systems of sine \(\left\{\sin nt\right\}_{n\geqslant 1}\) and cosine \(\left\{\cos nt\right\}_{n\geqslant 0}\) functions form bases in \(G_{p),\rho}(0,\pi)\).
期刊介绍:
Moscow University Mathematics Bulletin is the journal of scientific publications reflecting the most important areas of mathematical studies at Lomonosov Moscow State University. The journal covers research in theory of functions, functional analysis, algebra, geometry, topology, ordinary and partial differential equations, probability theory, stochastic processes, mathematical statistics, optimal control, number theory, mathematical logic, theory of algorithms, discrete mathematics and computational mathematics.