= Contour-shift proof of the Paley–Wiener–Schwartz theorem
For the forward implication, finite <order of a distribution> makes $F(\zeta)=\langle u,e^{-ix\cdot\zeta}\rangle$ entire. Use a smooth cutoff in an $\varepsilon$-neighborhood of the support set, with derivatives of order $j$ bounded by $C_j\varepsilon^{-j}$. Taking $\varepsilon=(1+|\zeta|)^{-1}$ in the finite-order estimate gives the polynomial factor and adds at most $e$ to the desired exponential bound. This shrinking cutoff recovers the exact support indicator, instead of an arbitrarily enlarged one.
For the converse, the real restriction of $F$ has <polynomial growth> and defines an inverse <tempered distribution>. If a <test function> $\varphi$ is supported in $x\cdot\omega\geq R+\eta$ for a unit vector $\omega$ and $\eta>0$, <contour shifting> gives
$$
\langle u,\varphi\rangle=(2\pi)^{-n}\int F(\xi+it\omega)\widehat\varphi(-\xi-it\omega)\,d\xi.
$$
Repeated <integration by parts> gives, for every integer $M$,
$$
|\widehat\varphi(-\xi-it\omega)|
\leq C_M(1+t)^{2M}e^{-(R+\eta)t}(1+|\xi|^2)^{-M}.
$$
Choosing $2M>N+n$ justifies the shift by <Cauchy integral theorem> and bounds the pairing by $C'(1+t)^{N+2M}e^{-\eta t}$, which tends to zero. Half-spaces of this form cover the complement of the ball, and a <partition of unity> proves the support inclusion. For general compact convex $K$, replace $R$ by $H_K(\omega)$ in each separating direction. <Fourier inversion> gives uniqueness. A related proof outline appears in https://math.mit.edu/~rbm/Problems4.pdf[Richard Melrose's distribution-theory problem set].
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