= 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 $\phi\in\mathcal D(0,1)$, the displayed estimate holds for every integer $N\geq1$. Let $h_N=P^Nf$ use <periodic primitives with zero mean>. Repeated <integration by parts> gives
$$
\int_0^1\overline f(jx)\phi(x)\,dx
=j^{-N}\int_0^1\overline h_N(jx)\phi^{(N)}(x)\,dx.
$$
There are no <boundary terms>, and integration over exactly $j$ periods gives $\|\overline h_N(j\,\cdot)\|_{L^1(0,1)}=\|h_N\|_{L^1}\leq\|f\|_{L^1}$. This proves the estimate. In particular, if zero-mean $g_j$ satisfy $\|g_j\|_{L^1}=O(j^m)$, their <periodic extensions> satisfy $\overline g_j(j\,\cdot)\to0$ in <weak convergence of distributions> by choosing $N=m+1$.
``
Target: Essential closure. Parent: Lebesgue measure. Intended file: measure-theory.bigb.
``
Back to article page
