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Let p≥1,ℓ∈N,α,β>-1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p\ge 1, \ell \in \mathbb {N}, \alpha ,\beta >-1$$\end{document} and ϖ=(ω0,ω1,…,ωℓ-1)∈Rℓ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varpi =(\omega _0,\omega _1, \ldots , \omega _{\ell -1})\in \mathbb {R}^{\ell }$$\end{document}. Given a suitable function f, we define the discrete–continuous Jacobi–Sobolev norm of f as: ‖f‖s,p:=∑k=0ℓ-1f(k)(ωk)p+∫-11f(ℓ)(x)pdμα,β(x)1p,\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} \Vert f \Vert _{{\scriptscriptstyle \mathsf {s}},{\scriptscriptstyle p}}:= \left( \sum _{k=0}^{\ell -1} \left| f^{(k)}(\omega _{k})\right| ^{p} + \int _{-1}^{1} \left| f^{(\ell )}(x)\right| ^{p} \mathrm{d}\mu ^{\alpha ,\beta }(x)\right) ^{\frac{1}{p}}, \end{aligned}$$\end{document}where dμα,β(x)=(1-x)α(1+x)βdx\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ \mathrm{d}\mu ^{\alpha ,\beta }(x)=(1-x)^{\alpha } (1+x)^{\beta }\mathrm{d}x$$\end{document}. Obviously, ‖·‖s,2=⟨·,·⟩s\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Vert \cdot \Vert _{{\scriptscriptstyle \mathsf {s}},{\scriptscriptstyle 2}}= \sqrt{\langle \cdot ,\cdot \rangle _{{\scriptscriptstyle \mathsf {s}}}}$$\end{document}, where ⟨·,·⟩s\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\langle \cdot ,\cdot \rangle _{{\scriptscriptstyle \mathsf {s}}}$$\end{document} is the inner product ⟨f,g⟩s:=∑k=0ℓ-1f(k)(ωk)g(k)(ωk)+∫-11f(ℓ)(x)g(ℓ)(x)dμα,β(x).\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} \langle f,g \rangle _{{\scriptscriptstyle \mathsf {s}}}:= \sum _{k=0}^{\ell -1} f^{(k)}(\omega _{k}) \, g^{(k)}(\omega _{k}) + \int _{-1}^{1} f^{(\ell )}(x) \,g^{(\ell )}(x) \mathrm{d}\mu ^{\alpha ,\beta }(x). \end{aligned}$$\end{document}In this paper, we summarize the main advances on the convergence of the Fourier–Sobolev series, in norms of type Lp\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^p$$\end{document}, in the continuous and discrete cases, respectively. Additionally, we study the completeness of the Sobolev space of functions associated with the norm ‖·‖s,p\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Vert \cdot \Vert _{{\scriptscriptstyle \mathsf {s}},{\scriptscriptstyle p}}$$\end{document} and the denseness of the polynomials. Furthermore, we obtain the conditions for the convergence in ‖·‖s,p\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Vert \cdot \Vert _{{\scriptscriptstyle \mathsf {s}},{\scriptscriptstyle p}}$$\end{document} norm of the partial sum of the Fourier–Sobolev series of orthogonal polynomials with respect to ⟨·,·⟩s\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\langle \cdot ,\cdot \rangle _{{\scriptscriptstyle \mathsf {s}}}$$\end{document}.
Bulletin of the Malaysian Mathematical Sciences Society – Springer Journals
Published: Jul 2, 2020
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