We prove an extension of Hardy’s classical characterization of real Gaussians of the form e−παx2, α > 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 ƒ and its Fourier transform ƒ̂ along some pair of lines in the complex plane is shown to imply that ƒ is a complex Gaussian.
Proceedings of the American Mathematical Society Vol. 134, Issue 5, p. 1459 - 1466