Hermite expansions and Hardy's theorem

Assuming that both a function and its Fourier transform are dominated by a Gaussian of large variance, it is shown that the Hermite coefficients of the function decay exponentially. A sharp estimate for the rate of exponential decay is obtained in terms of the variance, and in the limiting case (when the variance becomes so small that the Gaussian is its own Fourier transform), Hardy's theorem on Fourier transform pairs is obtained. A quantitative result on the confinement of particle-like states of a quantum harmonic oscillator is obtained. A stronger form of the result is conjectured. Further, it is shown how Hardy's theorem may be derived from a weak version of confinement without using complex analysis.