We establish Hermite expansion characterizations for several subspaces of the Fréchet space of functions on the real line satisfying |f(x)|≲e−(1/2−λ)x², |fˆ(ξ)|≲e−(1/2−λ)ξ², ∀λ>0. In particular, we extend and improve Fourier characterizations of the so-called proper Pilipović spaces obtained in [21]. The main ingredients in our proofs are the Bargmann transform and some achieved optimal forms of the Phragmén-Lindelöf principle.