  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.
