Uncertainty principle in Fourier analysis

Remarks

If $f$ vanishes outside a ball of radius $R$, then the integral $\int f(x) e^{-2\pi i x \cdot z}\, dx$ makes sense for every $z \in \C^n$. It defines an entire function of $z$ that extends $\hat f$. An entire function that vanishes on a nonempty open subset of $\R^n$ vanishes identically, so $\hat f = 0$, and then $f = 0$ by injectivity of the transform.

The statement is the simplest of a family of results that say a nonzero function and its Fourier transform cannot both be sharply concentrated. The Heisenberg inequality bounds the product of the variances of $\abs{f}^2$ and $\abs{\hat f}^2$ from below. Hardy's uncertainty principle is about Gaussian decay, and the entropic uncertainty principle measures spread by entropy. Benedicks' theorem replaces bounded sets by sets of finite measure.

Example: $\operatorname{rect}$, the indicator of $[-\frac12, \frac12]$, has compact support, and its transform $\sin(\pi\xi)/(\pi\xi)$ has zeros only at the nonzero integers, so it does not vanish outside any bounded set.

Going deeper

For a student

The proof uses the Paley–Wiener idea that compact support in $x$ makes $\hat f$ analytic, indeed entire of exponential type. Analyticity is very rigid, and this rigidity is the source of all qualitative uncertainty principles. In signal processing the statement says that no nonzero band-limited signal has finite duration, so every real filter or sampling scheme is a compromise between the two.