  We analyze why the discretization of linear transport with asymmetric Hermite
basis functions can be instable in quadratic norm. The main reason is that the
finite truncation of the infinite moment linear system looses the skew-symmetry
property with respect to the Gram matrix. Then we propose an original closed
formula for the scalar product of any pair of asymmetric basis functions. It
makes possible the construction of two simple modifications of the linear
systems which recover the skew-symmetry property. By construction the new
methods are quadratically stable with respect to the natural $L^2$ norm. We
explain how to generalize to other transport equations encountered in numerical
plasma physics. Basic numerical tests illustrate the unconditional stability
properties of our algorithms.
