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

We establish Hermite expansion characterizations for several subspaces of the Fr\'{e}chet space of functions on the real line satisfying the time-frequency decay bounds \[ |f(x)| e^{-(1/2 - λ) x²} , |{f}(ξ)| e^{-(1/2 - λ) ξ²} , ∀ λ > 0 . \] In particular, our results improve and extend upon recent Fourier characterizations of the so-called proper Pilpovi\'{c} 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 Phragm\'{e}n-Lindel\"{o}f principle on sectors, also provided in this article.