# HARDY’S THEOREM AND ROTATIONS

Hogan, Jeffrey A. and Lakey, Joseph D. (2006) HARDY’S THEOREM AND ROTATIONS. Proceedings of the American Mathematical Society, 134 (5). pp. 1459-1466. ISSN 0002-9939

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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 α 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 $\hat{f}$ along some pair of lines in the complex plane is shown to imply that $f$ is a complex Gaussian.