On the Hermite expansions of functions from Hardy class

Considering functions $f$ on $\R^n$ for which both $f$ and $\hat{f}$ are bounded by the Gaussian $e^{-{1/2}a|x|^2}, 0 < a < 1$ we show that their Fourier-Hermite coefficients have exponential decay. Optimal decay is obtained for $O(n)-$finite functions thus extending the one dimensional result of Vemuri.