= Rapid distributional decay of periodic oscillations
{title2=$\left|\int_0^1\overline f(jx)\phi(x)\,dx\right|\leq j^{-N}\|f\|_1\|\phi^{(N)}\|_\infty$}
If $f\in L^1(0,1)$ has mean zero, $j$ is a positive integer and $\phi\in\mathcal D(0,1)$, then for every integer $N\geq1$,
$$
\left|\int_0^1\overline f(jx)\phi(x)\,dx\right|\leq j^{-N}\|f\|_1\|\phi^{(N)}\|_\infty.
$$
Here the overline denotes the <periodic extension>, not complex conjugation. Let $h_N=P^Nf$ use <periodic primitives with zero mean>. Repeated <integration by parts> gives the signed identity
$$
\int_0^1\overline f(jx)\phi(x)\,dx=(-1)^Nj^{-N}\int_0^1\overline h_N(jx)\phi^{(N)}(x)\,dx.
$$
There are no <boundary terms>, and integration over exactly $j$ periods gives $\|\overline h_N(j\,\cdot)\|_1=\|h_N\|_1\leq\|f\|_1$. This proves the estimate. If zero-mean $g_j$ satisfy $\|g_j\|_1=O(j^m)$ for fixed real $m\geq0$, choose an integer $N>m$ to obtain $\overline g_j(j\,\cdot)\to0$ in <weak convergence of distributions>.
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