Benedicks' theorem

Remarks

The theorem strengthens the qualitative uncertainty principle, which assumes bounded sets instead of sets of finite measure. With finite measure only, the analytic continuation argument is no longer available. The proof periodizes instead. After a dilation one may assume that the set where $f \ne 0$ has measure less than $1$. For almost every $\xi$, the $\Z^n$-periodization of $x \mapsto f(x) e^{-2\pi i x \cdot \xi}$ has the numbers $\hat f(\xi + k)$, $k \in \Z^n$, as Fourier coefficients, as in the Poisson summation formula, and only finitely many of them are nonzero. So this periodization is a trigonometric polynomial, and it vanishes on a set of positive measure in the torus. A trigonometric polynomial that vanishes on a set of positive measure is zero, so $\hat f(\xi + k) = 0$ for all $k$ and almost every $\xi$, and $f = 0$.

The theorem is due to M. Benedicks. A version for $L^2(\R^n)$ was proved independently by W. O. Amrein and A. M. Berthier.

Example: the Gaussian $e^{-\pi \abs{x}^2}$ is nowhere zero, as is its transform, so both sets have infinite measure, consistent with the theorem.

Going deeper

For a student

This is the measure-theoretic form of the principle that $f$ and $\hat f$ cannot both be concentrated. A quantitative version, due to Nazarov in dimension one and extended to $\R^n$, bounds $\norm{f}_2$ by a constant times the $L^2$ norms of $f$ and $\hat f$ outside such sets. A pair of sets with this property is called an annihilating pair, and the theorem says that any two sets of finite measure form one.