  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 \[ |f(x)| \lesssim e^{-(\frac{1}{2} - \lambda) x^{2}} , \qquad
|\widehat{f}(\xi)| \lesssim e^{-(\frac{1}{2} - \lambda) \xi^{2}} , \qquad
\forall \lambda > 0 . \] 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.
