  We prove the convergence of a spectral discretization of the Vlasov-Poisson
system. The velocity term of the Vlasov equation is discretized using either
Hermite functions on the infinite domain or Legendre polynomials on a bounded
domain. The spatial term of the Vlasov and Poisson equations is discretized
using periodic Fourier expansions. Boundary conditions are treated in weak form
through a penalty type term, that can be applied also in the Hermite case. As a
matter of fact, stability properties of the approximated scheme descend from
this added term. The convergence analysis is carried out in details for the
1D-1V case, but results can be generalized to multidimensional domains,
obtained as Cartesian product, in both space and velocity. The error estimates
show the spectral convergence, under suitable regularity assumptions on the
exact solution.
