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

We establish Hermite expansion characterizations for several subspaces of the Fr\'echet space of functions on the real line satisfying the time-frequency decay bounds \ |f (x) | e^- (1{2 -) x^2}, |f () | e^- (1{2 -) ^2}, > 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\'en-Lindel\"of principle on sectors, also provided in this article.