= Solution
Let $M=\sup_j j^{-m}\|g_j\|_{L^1}<\infty$. For each $j$, apply the <periodic primitives with zero mean> construction of part (iii) to $g_j$, writing $h_{j,0}=g_j$ and
$$
h_{j,r+1}=Rh_{j,r}-\int_0^1Rh_{j,r}.
$$
The estimates are independent of $j$: $\|h_{j,r}\|_{L^1}\leq\|g_j\|_{L^1}$. With $N=m+1$, the same repeated <integration by parts> gives, for every <test function> $\phi\in\mathcal D(0,1)$,
$$
\begin{aligned}
|\langle u_j,\phi\rangle|
&=j^{-N}\left|\int_0^1\overline h_{j,N}(jx)\phi^{(N)}(x)\,dx\right|\\
&\leq j^{-m-1}\|g_j\|_{L^1}\|\phi^{(m+1)}\|_\infty
\leq\frac{M}{j}\|\phi^{(m+1)}\|_\infty\longrightarrow0.
\end{aligned}
$$
This is precisely <weak convergence of distributions>, so $\boxed{u_j\to0\text{ in }\mathcal D'(0,1)}$. The extra <integration by parts> beyond the growth exponent $m$ is what ensures decay.
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