Fourier uncertainty principles

Abstract

A mathematical uncertainty principle is an inequality or uniqueness theorem concerning the joint localization of a function or system and its spectrum. A Fourier uncertainty principle is thus a statement that a function and its Fourier transform cannot both decay too rapidly. The most familiar form is the Wiener-Heisenberg inequality,

$$ \left\| {(x - x_0 )f(x)} \right\|_2 \left\| {(\xi - \xi _0 )\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{f} (\xi )} \right\|_2 \geqslant \frac{{\mathop {\left\| f \right\|}\nolimits_2^2 }} {{4\pi }},$$

(5.1)

which becomes an equality only for multiples of Gaussians of the form exp(−πα(x − x₀)² + 2πiξ₀x). In this chapter, D will denote the normalized differentiation or momentum operator D = (1/2πi)d/dx. The inequality (5.1) is often written

$$ \sigma _X (f)\sigma _D (f) \equiv \left\| {Xf - \left\langle {Xf,f} \right\rangle f} \right\|_2 \left\| {Df - \left\langle {Df,f} \right\rangle f} \right\|_2 \geqslant 1,$$

(5.2)

where X is the multiplication operator f(x) ↦ xf(x). Barnes [22] attributes the first proof of (5.1) to a lecture delivered by Wiener in Hilbert’s mathematical physics seminar in Göttingen in 1925. In fact, Wiener discusses this lecture, though not the explicit proof of (5.1), at length in his autobiography (see [365], p. 105). What distinguished Wiener’s approach, however, was his macroscopic interpretation of (5.1), a point of view that led to further important achievements, including the Landau, Pollack and Slepian “Bell Labs” uncertainty principles in Chapter 3