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$\left|f\right(x\left)\right|\lesssim e^{-(\frac{1}{2}-\lambda )x^2},\left|\hat{f}\right(\xi \left)\right|\lesssim e^{-(\frac{1}{2}-\lambda ){\xi }^2},\forall \lambda >0.$ 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.