Discrete-Continuous Jacobi-Sobolev Spaces and Fourier Series

Let $p\geq 1$, $\ell\in \NN$, $\alpha,\beta>-1$ and $\varpi=(\omega_0,\omega_1, \dots, \omega_{\ell-1})\in \RR^{\ell}$. Given a suitable function $f$, we define the discrete-continuous Jacobi-Sobolev norm of $f$ as: $$\normSp{f}:= \left(\sum_{k=0}^{\ell-1} \left|f^{(k)}(\omega_{k})\right|^{p} + \int_{-1}^{1} \left|f^{(\ell)}(x)\right|^{p} d\Jm(x)\right)^{\frac{1}{p}},$$ where $d\Jm(x)=(1-x)^{\alpha} (1+x)^{\beta}dx$. Obviously, $\normSp[2]{\cdot}= \sqrt{\IpS{\cdot}{\cdot}}$, where $\IpS{\cdot}{\cdot}$ is the inner product. $$\IpS{f}{g}:= \sum_{k=0}^{\ell-1} f^{(k)}(\omega_{k}) \, g^{(k)}(\omega_{k}) + \int_{-1}^{1} f^{(\ell)}(x) \,g^{(\ell)}(x) d\Jm(x).$$ 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 of functions associated with the norm $\normSp{\cdot}$ and the denseness of the polynomials. Furthermore, we obtain the conditions for the convergence in $\normSp{\cdot}$ norm of the partial sum of the Fourier-Sobolev series of orthogonal polynomials with respect to $\IpS{\cdot}{\cdot}$ .