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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 theirFourier-Hermite coefficients have exponential decay. Optimal decay is obtainedfor $ O(n)-$finite functions thus extending the one dimensional result ofVemuri.

On the Hermite expansions of functions from Hardy class