= Paley–Wiener–Schwartz theorem
{c}
For a compact convex set $K\subset\mathbb R^n$, the <Fourier transform> gives a bijection between distributions supported in $K$ and entire functions satisfying
$$
|F(\zeta)|\leq C(1+|\zeta|)^N e^{H_K(\operatorname{Im}\zeta)},
\qquad H_K(\eta)=\sup_{x\in K}x\cdot\eta,
$$
for some $C,N$. Here $H_K$ is the <support function>. In the closed-ball case $K=\overline B_R$, the exponential factor is $e^{R|\operatorname{Im}\zeta|}$. Convexity is essential to infer support in $K$ from this bound: the support function cannot distinguish a nonconvex set from its convex hull.
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