Bounded Fourier support

Definition

A function $f\in L^2(\mathbb R^d)$ has bounded Fourier support if its $L^2$ Fourier transform vanishes almost everywhere outside some bounded set $K\subset\mathbb R^d$ :

\operatorname{supp}\widehat f\subseteq K.

When $K=B_\sigma$ , the number $\sigma$ is a frequency or spectral radius.

Meaning for an

L^2

function

The Fourier transform is defined through the Plancherel theorem, so its support is the essential support of an $L^2$ -equivalence class. Altering $\widehat f$ on a null set does not change the condition.

Analytic characterization

The Paley–Wiener theorem identifies ball-supported Fourier transforms with restrictions of entire functions of several variables of controlled exponential type. Bounded Fourier support is therefore a strong analyticity constraint, not merely a decay condition.

References

Lars Hörmander, The Analysis of Linear Partial Differential Operators I, Springer, 2003. DOI record. Relevant: §7.3.