Hardy’s Theorem and Rotations

We prove an extension of Hardy’s classical characterization of real Gaussians of the form e−παx 2 , 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 f and its Fourier transform f̂ along some pair of lines in the complex plane is shown to imply that f is a complex Gaussian. 1. Hardy’s theorem and Fourier Uncertainty A Fourier uncertainty principle is, generally speaking, a statement that limits the rate at which a function and its Fourier transform can decay, or otherwise restricts the “joint localization” of a Fourier pair. We normalize the Fourier transform f̂ of f by setting f̂(ξ) = ∫∞ −∞ f(x)e −2πixξ dξ when f ∈ L(R). Hardy’s theorem is an uncertainty principle stating that if |f(x)| ≤ Ce−παx2 and |f̂(ξ)| ≤ C ′e−πβξ2 then: (i) if αβ > 1 then f = 0, while (ii) if αβ = 1 then f is a multiple of e−παx 2 . Hardy’s theorem has been extended in several different directions in recent years, including extensions to Euclidean space (e.g., [SST], [THA]) and, much more generally, to groups of homogeneous type (e.g., [ACDS]) and semisimple Lie groups (e.g., [SS], [CSS], cf. also [S]), and to statements about decay of time-frequency distributions on phase space (e.g., [GZ], [BDJ], [GR], [HL]). Other important directions include generalizations of Beurling’s important variation of Hardy’s theorem ([H], [BDJ], [CP], [BR]) and statements about decay of eigenvalues of Hermite expansions and related operators (e.g., [JAVE], [HL2]). The insightful survey [FS] discusses several of these developments in the context of Fourier uncertainty principles. Nearly all of the extensions just noted are proved by reduction to Hardy’s original theorem (or to Beurling’s theorem). In any case, they all reduce to some form of the maximum principle. Given all these directions of generalization, it is interesting to ask whether any new light can be shed on the base case of localization of analytic functions and their Fourier transforms. Many years ago, Gelfand and Shilov [GS] proved mapping properties under the Fourier transform of analytic functions satisfying certain growth/decay conditions along R and iR. As Hörmander [H] pointed out, the Gelfand-Shilov estimates show that Beurling’s theorem and other variations and extensions of Hardy’s theorem are, Date: September 21, 2004. 2000 Mathematics Subject Classification. Primary 42A38; Secondary 30D15.