{"title":"Convergence of the Fourier Series in Meixner–Sobolev Polynomials and Approximation Properties of Its Partial Sums","authors":"R. M. Gadzhimirzaev","doi":"10.1134/s0001434624030027","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We study the convergence of Fourier series in the polynomial system <span>\\(\\{m_{n,N}^{\\alpha,r}(x)\\}\\)</span> orthonormal in the sense of Sobolev and generated by the system of modified Meixner polynomials. In particular, we show that the Fourier series of <span>\\(f\\in W^r_{l^p_{\\rho_N}(\\Omega_\\delta)}\\)</span> in this system converges to <span>\\(f\\)</span> pointwise on the grid <span>\\(\\Omega_\\delta\\)</span> as <span>\\(p\\ge2\\)</span>. In addition, we study the approximation properties of partial sums of Fourier series in the system <span>\\(\\{m_{n,N}^{0,r}(x)\\}\\)</span>. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-07-05","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/s0001434624030027","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
We study the convergence of Fourier series in the polynomial system \(\{m_{n,N}^{\alpha,r}(x)\}\) orthonormal in the sense of Sobolev and generated by the system of modified Meixner polynomials. In particular, we show that the Fourier series of \(f\in W^r_{l^p_{\rho_N}(\Omega_\delta)}\) in this system converges to \(f\) pointwise on the grid \(\Omega_\delta\) as \(p\ge2\). In addition, we study the approximation properties of partial sums of Fourier series in the system \(\{m_{n,N}^{0,r}(x)\}\).
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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