Discrete-Continuous Jacobi-Sobolev Spaces and Fourier Series.

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: $$\Vert f\Vert_{s,p} := \left(\sum_{k=0}^{\ell-1} \left|f^{(k)}(\omega_{k})\right|^{p} + \int_{-1}^{1} \left|f^{(\ell)}(x)\right|^{p} d\mu^{\alpha,\beta}(x)\right)^{\frac{1}{p}},$$ where $ d\mu^{\alpha,\beta}(x)=(1-x)^{\alpha} (1+x)^{\beta}dx$. Obviously, $\Vert\cdot\Vert_{s,2} = \sqrt{\langle\cdot, \cdot\rangle_s}$, where $\langle\cdot, \cdot\rangle_s$ is the inner product $$\langle f,g\rangle_s := \sum_{k=0}^{\ell-1} f^{(k)}(\omega_{k}) \, g^{(k)}(\omega_{k}) + \int_{-1}^{1} f^{(\ell)}(x) \,g^{(\ell)}(x) d\mu^{\alpha,\beta}(x).$$ In this paper, we summarize the main advances on the convergence of the Fourier-Sobolev series, in norms of type $L^p$, cases continuous and discrete. We study the completeness of the Sobolev space associated with the norm $\Vert\cdot\Vert_{s,p}$ and the density of the polynomials. Furthermore, we obtain the conditions for the convergence in $\Vert\cdot\Vert_{s,p}$ norm of the partial sum of the Fourier-Sobolev series of orthogonal polynomials with respect to $\langle\cdot, \cdot\rangle_s$.