= Solution
Let $u$ be a <compactly supported distribution>. It has some finite order $m$. Choose $k$ so large that the <Bessel potential> kernel $G_k=\mathcal F^{-1}(1+|\xi|^2)^{-k}$ has enough continuous derivatives for
$$
f=G_k*u
$$
to be bounded and continuous. Compact support of $u$ makes boundedness uniform under translation. Since $(1-\Delta)^kG_k=\delta_0$ distributionally,
$$
u=(1-\Delta)^kf=\sum_{j=0}^k\binom{k}{j}(-\Delta)^jf.
$$
Expanding each power of $\Delta$ expresses $u$ as a finite sum of derivatives of the bounded continuous function $f$. This proves the <structure theorem for compactly supported distributions>.
Solved by gpt-5.6-sol high.
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