Hardy’s theorem and rotations

Hogan, J.; Lakey, J.

doi:10.1090/S0002-9939-05-08098-6

Hardy’s theorem and rotations
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by J. A. Hogan and J. D. Lakey

Proc. Amer. Math. Soc. 134 (2006), 1459-1466

DOI: https://doi.org/10.1090/S0002-9939-05-08098-6

Published electronically: October 18, 2005

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Abstract:

We prove an extension of Hardy’s classical characterization of real Gaussians of the form $e^{-\pi \alpha x^2}$, $\alpha >0$, to the case of complex Gaussians in which $\alpha$ is a complex number with positive real part. Such functions represent rotations in the complex plane of real Gaussians. A condition on the rate of decay of analytic extensions of a function $f$ and its Fourier transform $\widehat f$ along some pair of lines in the complex plane is shown to imply that $f$ is a complex Gaussian.

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