 Paley–Wiener–Schwartz theorem

For a compact convex set $K\subset {\mathbb{R}}^n$, the Fourier transform gives a bijection between distributions supported in $K$ and entire functions satisfyingfor some $C,N$. Here $H_K$ is the support function. In the closed-ball case $K={\overline{B}}_R$, the exponential factor is $e^{R|\mathrm{Im}\zeta |}$. Convexity is essential to infer support in $K$ from this bound: the support function cannot distinguish a nonconvex set from its convex hull.