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Discrete-continuous Jacobi-Sobolev spaces and Fourier series. (English) Zbl 1462.42047

Summary: Let \(p\geq 1\), \(\ell\in\mathbb{N}\), \(\alpha,\beta>-1\) and \(\varpi=(\omega_0,\omega_1,\dots,\omega_{\ell-1})\in\mathbb{R}^{\ell}\). Given a suitable function \(f\), we define the discrete-continuous Jacobi-Sobolev norm of \(f\) as: \[ \Vert f\Vert_{\mathsf{s},p}:=\bigg(\sum\limits_{k=0}^{\ell-1}\left|f^{(k)}(\omega_k)\right|^p+\int_{-1}^1\left|f^{(\ell)}(x)\right|^p\text{d}\mu^{\alpha,\beta}(x)\bigg)^{\frac{1}{p}}, \] where \(\text{d}\mu^{\alpha,\beta}(x)=(1-x)^{\alpha}(1+x)^{\beta}\text{d}x\). Obviously, \(\Vert\cdot\Vert_{\mathsf{s},2}=\sqrt{\langle\cdot,\cdot\rangle_{\mathsf{s}}}\), where \(\langle\cdot,\cdot\rangle_{\mathsf{s}}\) is the inner product \[ \langle f,g\rangle_{\mathsf{s}}:=\sum\limits_{k=0}^{\ell-1}f^{(k)}(\omega_k)\, g^{(k)}(\omega_k)+\int_{-1}^1 f^{(\ell)}(x)\,g^{(\ell)}(x)\text{d}\mu^{\alpha,\beta}(x). \] 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 \(\Vert\cdot\Vert_{\mathsf{s},p}\) and the denseness of the polynomials. Furthermore, we obtain the conditions for the convergence in \(\Vert\cdot\Vert_{\mathsf{s},p}\) norm of the partial sum of the Fourier-Sobolev series of orthogonal polynomials with respect to \(\langle\cdot,\cdot\rangle_{\mathsf{s}}\).

MSC:

42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
42A20 Convergence and absolute convergence of Fourier and trigonometric series
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

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References:

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