Convergence of Hermite expansions in modulation spaces

The aim of this paper is to give an elementary proof that Hermite expensions of a function $f$ in the modulation space $M^p(R)$ converges to $f$ in $M^p(R)$ when $1< p<+\infty$ and may diverge when $p = 1,\infty$. The result was previously established for $1< p<+\infty$ by Garling and Wojtaszczyk and for $p = 1,\infty$ by Lusky in an equivalent setting of Fock spaces by different methods. Higher dimesional results are also considered. In an appendix, we also establish upper bounds for the Zak transform of Hermite functions.