  We consider a particular discretization of the harmonic oscillator which
admits an orthogonal basis of eigenfunctions called Kravchuk functions
possessing appealing properties from the numerical point of view. We
analytically prove the almost second-order convergence of these discrete
functions towards Hermite functions, uniformly for large numbers of modes. We
then describe an efficient way to simulate these eigenfunctions and the
corresponding transformation. We finally show some numerical experiments
corroborating our different results.
