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Hermite expansions and Hardy's theorem | | M. K. Vemuri
; | | Date: |
15 Jan 2008 | | Abstract: | 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. | | Source: | arXiv, 0801.2234 | | Services: | Forum | Review | PDF | Favorites | |
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