Paley–Wiener theorem for bounded Fourier support

Statement

Let $f\in L^2(\mathbb R^d)$ and $\sigma>0$ . With the Fourier convention $\widehat f(\xi)=\int f(x)e^{-2\pi i x\cdot\xi}\,dx$ , the condition

\operatorname{supp}\widehat f\subseteq B_{\sigma/(2\pi)}

holds if and only if $f$ is the restriction to $\mathbb R^d$ of an entire function $F:\mathbb C^d\to\mathbb C$ for which

|F(x+iy)|\le A e^{\sigma|y|}

for some $A>0$ and all $x,y\in\mathbb R^d$ .

Forward direction

Fourier inversion over the bounded support defines $F(z)=\int\widehat f(\xi)e^{2\pi i z\cdot\xi}\,d\xi$ . The bounded frequency region makes the integral entire and gives the exponential bound by Cauchy–Schwarz.

Reverse direction

Shift the contour in a complex line parallel to a fixed frequency $\xi$ . When $2\pi|\xi|>\sigma$ , exponential decay in the shifted half-plane forces $\widehat f(\xi)=0$ . A standard approximation removes auxiliary Schwartz regularity used in the contour argument.

Normalization warning

If the Fourier exponential is $e^{-ix\cdot\xi}$ , the ball radius is $\sigma$ rather than $\sigma/(2\pi)$ .

References

Lars Hörmander, The Analysis of Linear Partial Differential Operators I, Springer, 2003. DOI record. Relevant: Theorem 7.3.1.
Alex Cohen, “Fractal uncertainty in higher dimensions,” 2024. arXiv record. Relevant: Theorem 2.1 and Appendix A.3.