 Solution

Let $M={\sup }_j{j}^{-m}\Vert g_j{\Vert }_{L^1}<\infty$. For each $j$, apply the periodic primitives with zero mean construction of part (iii) to $g_j$, writing $h_{j,0}=g_j$ andThe estimates are independent of $j$: $\Vert h_{j,r}{\Vert }_{L^1}\le \Vert g_j{\Vert }_{L^1}$. With $N=m+1$, the same repeated integration by parts gives, for every test function $\varphi \in \mathcal{D}(0,1)$,This is precisely weak convergence of distributions, so $\boxed{u_j\to 0\text{ in }{\mathcal{D}}^{\prime }(0,1)}$. The extra integration by parts beyond the growth exponent $m$ is what ensures decay.