Hardy's uncertainty principle

Remarks

By the Fourier transform of the Gaussian, $e^{-\pi a x^2}$ has transform $a^{-1/2} e^{-\pi \xi^2 / a}$, so case 2 cannot be improved. If $ab < 1$, then $e^{-\pi c x^2}$ satisfies both bounds, with a suitable constant $C$, for every $c$ with $a \le c \le 1/b$, so there are nonzero functions of this kind.

The proof uses complex analysis. The decay of $f$ makes $\hat f$ extend to an entire function with $\abs{\hat f(\xi + i\eta)} \le C' e^{\pi \eta^2 / a}$. In case 2, the entire function $g(z) = e^{\pi z^2 / a} \hat f(z)$ is bounded on the real and on the imaginary axis. The Phragmén–Lindelöf principle, applied in each quadrant, shows that $g$ is bounded, hence constant by Liouville's theorem, and Fourier inversion gives $f$. Case 1 reduces to case 2 with $b$ replaced by $1/a$, and then the bound $\abs{\hat f(\xi)} \le C e^{-\pi b \xi^2}$ with $b > 1/a$ forces the constant to be $0$.

The result is due to G. H. Hardy. There are versions on $\R^n$, and versions with $L^p$ bounds in place of pointwise bounds.

Going deeper

For a student

Unlike the Heisenberg inequality, which measures spread with variances, Hardy's theorem measures concentration through pointwise decay. It is a sharp qualitative statement, and the borderline case singles out the Gaussians, just as the equality case of Heisenberg's inequality does. Its proof is a model for the use of Phragmén–Lindelöf arguments in Fourier analysis. It has been extended to Lie groups and to the Schrödinger equation, where it gives unique continuation results.