Hermite expansions for spaces of functions with nearly optimal time-frequency decay

We establish Hermite expansion characterizations for several subspaces of the Fréchet space of functions on the real line satisfying the time-frequency decay bounds |f(x)| ≲ e^-(1/2 - λ) x^2 , |f(ξ)| ≲ e^-(1/2 - λ) ξ^2 , ∀λ > 0 . In particular, our results improve and extend upon recent Fourier characterizations of the so-called proper Pilpović spaces obtained in [J. Funct. Anal. 284 (2023), 109724]. The main ingredients in our proofs are the Bargmann transform and some optimal forms of the Phragmén-Lindelöf principle on sectors, also provided in this article.