Bulletin of the Malaysian Mathematical Sciences Society ( IF 1.0 ) Pub Date : 2020-07-02 , DOI: 10.1007/s40840-020-00950-7 Abel Díaz-González , Francisco Marcellán , Héctor Pijeira-Cabrera , Wilfredo Urbina
Let \(p\ge 1, \ell \in \mathbb {N}, \alpha ,\beta >-1\) and \(\varpi =(\omega _0,\omega _1, \ldots , \omega _{\ell -1})\in \mathbb {R}^{\ell }\). Given a suitable function f, we define the discrete–continuous Jacobi–Sobolev norm of f as:
$$\begin{aligned} \Vert f \Vert _{{\scriptscriptstyle \mathsf {s}},{\scriptscriptstyle p}}:= \left( \sum _{k=0}^{\ell -1} \left| f^{(k)}(\omega _{k})\right| ^{p} + \int _{-1}^{1} \left| f^{(\ell )}(x)\right| ^{p} \mathrm{d}\mu ^{\alpha ,\beta }(x)\right) ^{\frac{1}{p}}, \end{aligned}$$where \( \mathrm{d}\mu ^{\alpha ,\beta }(x)=(1-x)^{\alpha } (1+x)^{\beta }\mathrm{d}x\). Obviously, \(\Vert \cdot \Vert _{{\scriptscriptstyle \mathsf {s}},{\scriptscriptstyle 2}}= \sqrt{\langle \cdot ,\cdot \rangle _{{\scriptscriptstyle \mathsf {s}}}}\), where \(\langle \cdot ,\cdot \rangle _{{\scriptscriptstyle \mathsf {s}}}\) is the inner product
$$\begin{aligned} \langle f,g \rangle _{{\scriptscriptstyle \mathsf {s}}}:= \sum _{k=0}^{\ell -1} f^{(k)}(\omega _{k}) \, g^{(k)}(\omega _{k}) + \int _{-1}^{1} f^{(\ell )}(x) \,g^{(\ell )}(x) \mathrm{d}\mu ^{\alpha ,\beta }(x). \end{aligned}$$In this paper, we summarize the main advances on the convergence of the Fourier–Sobolev series, in norms of type \(L^p\), in the continuous and discrete cases, respectively. Additionally, we study the completeness of the Sobolev space of functions associated with the norm \(\Vert \cdot \Vert _{{\scriptscriptstyle \mathsf {s}},{\scriptscriptstyle p}}\) and the denseness of the polynomials. Furthermore, we obtain the conditions for the convergence in \(\Vert \cdot \Vert _{{\scriptscriptstyle \mathsf {s}},{\scriptscriptstyle p}}\) norm of the partial sum of the Fourier–Sobolev series of orthogonal polynomials with respect to \(\langle \cdot ,\cdot \rangle _{{\scriptscriptstyle \mathsf {s}}}\).
中文翻译：
离散–连续Jacobi–Sobolev空间和傅立叶级数
令\（p \ ge 1，\ ell \ in \ mathbb {N}，\ alpha，\ beta> -1 \）和\（\ varpi =（\ omega _0，\ omega _1，\ ldots，\ omega _ { \ ell -1}）\ in \ mathbb {R} ^ {\ ell} \）中。给定合适的函数˚F，我们定义的离散连续雅克比的Sobolev范数˚F为：
$$ \ begin {aligned} \ Vert f \ Vert _ {{\ scriptscriptstyle \ mathsf {s}}，{\ scriptstyle p}}：= \ left（\ sum _ {k = 0} ^ {\ ell -1} \ left | f ^ {（（k）}（\ omega _ {k}）\ right | ^ {p} + \ int _ {-1} ^ {1} \ left | f ^ {（\ ell）}（x ）\ right | ^ {p} \ mathrm {d} \ mu ^ {\ alpha，\ beta}（x）\ right）^ {\ frac {1} {p}}，\ end {aligned} $$其中\（\ mathrm {d} \ mu ^ {\ alpha，\ beta}（x）=（1-x）^ {\ alpha}（1 + x）^ {\ beta} \ mathrm {d} x \）。显然，\（\ Vert \ cdot \ Vert _ {{\ scriptscriptstyle \ mathsf {s}}，{\ scriptscriptstyle 2}} = \ sqrt {\ langle \ cdot，\ cdot \ rangle _ {{\ scriptscriptstyle \ mathsf {s }}}} \），其中\（\ langle \ cdot，\ cdot \ rangle _ {{\ scriptscriptstyle \ mathsf {s}}} \）是内积
$$ \ begin {aligned} \ langle f，g \ rangle _ {{\ scriptscriptstyle \ mathsf {s}}}：= \ sum _ {k = 0} ^ {\ ell -1} f ^ {（k）} （\ omega _ {k}）\，g ^ {（k）}（\ omega _ {k}）+ \ int _ {-1} ^ {1} f ^ {（\ ell}}（x）\， g ^ {（\ ell）}（x）\ mathrm {d} \ mu ^ {\ alpha，\ beta}（x）。\ end {aligned} $$在本文中，我们总结了分别在连续和离散情况下在（（L ^ p \）类型的范数下）的Fourier–Sobolev级数收敛的主要进展。此外，我们研究了与范数\（\ Vert \ cdot \ Vert _ {{\ scriptscriptstyle \ mathsf {s}}，{\ scriptscriptstyle p}} \）相关的函数的Sobolev空间的完备性和多项式的密度。此外，我们获得了傅立叶-Sobolev级数的部分和的\（\ Vert \ cdot \ Vert _ {{\ scriptscriptstyle \ mathsf {s}}，{\ scriptscriptstyle p}} \）范式中的收敛条件。关于\（\ langle \ cdot，\ cdot \ rangle _ {{\\ scriptscriptstyle \ mathsf {s}}} \）的正交多项式。
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