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Hardy's theorem and rotations

Author(s): J. A. Hogan; J. D. Lakey
Journal: Proc. Amer. Math. Soc. 134 (2006), 1459-1466.
MSC (2000): Primary 42A38; Secondary 30D15
Posted: 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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Additional Information:

J. A. Hogan
Affiliation: Department of Mathematical Sciences, University of Arkansas, Fayetteville, Arkansas 72701
Email: jeffh@uark.edu

J. D. Lakey
Affiliation: Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003--8001
Email: jlakey@nmsu.edu

DOI: 10.1090/S0002-9939-05-08098-6
PII: S 0002-9939(05)08098-6
Keywords: Hardy's theorem, uncertainty principle
Received by editor(s): September 24, 2004
Received by editor(s) in revised form: December 20, 2004
Posted: October 18, 2005
Additional Notes: This research was supported by a Macquarie University MURG grant
Communicated by: Juha M. Heinonen
Copyright of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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