Stability of Semi-Lagrangian schemes of arbitrary odd degree under constant and variable advection speed

The equivalence between semi-Lagrangian and Lagrange-Galerkin schemes has been proved by R. Ferretti [J. Comp. Math. 28 (2010), no. 4, 461-473], [Numerische Mathematik 124 (2012), no. 1, 31-56] for the case of centered Lagrange interpolation of odd degree p\le 13. We generalize this result to an arbitrary odd degree, for both the case of constant- and variable-coefficient equations. In addition, we prove that the same holds for spline interpolations.