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 \begin{equation*} |f(x)| \lesssim e^{-(\frac{1}{2} - \lambda ) x^{2}} , \qquad | \widehat{f}(\xi )| \lesssim e^{-(\frac{1}{2} - \lambda ) \xi ^{2}} , \qquad \forall \lambda > 0 . \end{equation*} In particular, we extend and improve Fourier characterizations of the so-called proper Pilipovi\'{c} spaces obtained in [J. Funct. Anal. 284 (2023), 109724]. The main ingredients in our proofs are the Bargmann transform and some achieved optimal forms of the Phragm\'{e}n-Lindel\"{o}f principle.