Convergence of Hermite expansions in modulation spaces

The aim of this paper is to give an elementary proof that the Hermite expansion of a function in the modulation space converges to in when and may diverge when , or when . Although related results exist in the context of Fock spaces, their formulation and consequences for modulation spaces appear to have gone unnoticed in time–frequency analysis. More precisely, the result was previously established for by Garling and Wojtaszczyk, for by Lusky, and for by Li in the equivalent setting of Fock spaces, using different methods. Higher dimensional results are also considered.