Hardy’s theorem and rotations

We prove an extension of Hardy’s classical characterization of real Gaussians of the form e − π α x 2 e^{-\pi \alpha x^2} , α > 0 \alpha >0 , to the case of complex Gaussians in which α \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 f and its Fourier transform f ^ \widehat f along some pair of lines in the complex plane is shown to imply that f f is a complex Gaussian.