We consider functions $f$ on $\mathbb{R}^n$ for which both $f$ and their Fourier transforms $\hat{f}$ are bounded by the Gaussian
 $e^{-\frac{a}{2}|x|^2}$ for some $0. Using the Bargmann transform, we show that their Fourier-Hermite coefficients have exponential decay. 
 This is an extension of the one dimensional result of M. K. Vemuri, in whichsharp estimates were proved. In higher dimensions, we obtain the 
 analogous result for functions $f$ which are $O(n)$-finite. Here by an $O(n)$-finite function we mean a function whose restriction to the unit 
 sphere $S^{n-1}$ has only finitely many terms in its spherical harmonic expansion. Some partial results are proved for general functions. As a 
 corollary to these results, we obtain Hardy's uncertainty principle. An analogous problem is studied in the case of Beurling's uncertainty principle.
 Next we consider the structure of analytic and entire vectors for the Schrodinger representations of the Heisenberg group. Using refined versions of 
 Hardy's theorem and their connection with Hermite expansions we obtain very precise representation theorems for analytic and entire vectors.