Sobolev Orthogonal Systems with Two Discrete Points and Fourier Series

被引：0
作者：
M. G. Magomed-Kasumov
机构：
[1] Daghestan Federal Research Centre of the Russian Academy of Sciences,
[2] Southern Mathematical Institute of the Vladikavkaz Scientific Center of Russian Academy of Sciences,undefined
来源：
Russian Mathematics | 2021年 / 65卷
关键词：
discrete-continuous inner product; Sobolev inner product; Faber–Schauder system; Jacobi polynomials with negative parameters; Fourier series; uniform convergence; coincidence at the ends of the segment; completeness of Sobolev systems;
D O I：
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学科分类号：
摘要：
We consider the properties of systems of functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Phi_1$\end{document} orthogonal with respect to a discrete-continuous Sobolev inner product of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\langle f,g \rangle_S = f(a)g(a)+f(b)g(b)+\int_a^b f'(t)g'(t)dt$\end{document}. In particular, we study the completeness of systems \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Phi_1$\end{document} in the Sobolev space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$W^1_{L^2}$\end{document}. Additionally, we analyze the properties of Fourier series with respect to systems \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Phi_1$\end{document}, and prove that these series converge uniformly to functions from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$W^1_{L^2}$\end{document}.
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页码：47 / 55
页数：8
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