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 $\IpS{ \cdot}{ \cdot}$ .
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