---
tag: 4ZY2D6C8A6
kind: statement
role: theorem
name: Benedicks' theorem
msc: ["42B10", "42A38"]
uses: [DPVM34ADZV, SASP14MA1Z, 2Z2QVWZ262, S1FB2K14NV, GARGJYSGFG]
about: [DPVM34ADZV]
sources:
  - url: https://en.wikipedia.org/wiki/Fourier_transform
    locator: Uncertainty principle
---

## Statement

Let $n \ge 1$ and let $f \in L^1(\R^n)$.
Let $\hat f(\xi) = \int_{\R^n} f(x) e^{-2\pi i x \cdot \xi}\, dx$ be its [[Fourier transform|DPVM34ADZV]].
Suppose that the sets $\set{x \in \R^n : f(x) \ne 0}$ and $\set{\xi \in \R^n : \hat f(\xi) \ne 0}$ both have finite [[Lebesgue measure|2Z2QVWZ262]].
Then $f = 0$ almost everywhere.

## Remarks

The theorem strengthens the [[qualitative uncertainty principle|S1FB2K14NV]], 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|GARGJYSGFG]], 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.

## Explanations

### L1 human

A nonzero function cannot live on a set of finite total size while its spectrum also lives on a set of finite total size.
The sets may be scattered and unbounded; only their total size matters.

### L2 human

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.