 Rapid distributional decay of periodic oscillations

If $f\in L^1(0,1)$ has mean zero, $j$ is a positive integer, and $\varphi \in \mathcal{D}(0,1)$, the displayed estimate holds for every integer $N\ge 1$. Let $h_N=P^{N}f$ use periodic primitives with zero mean. Repeated integration by parts givesThere are no boundary terms, and integration over exactly $j$ periods gives $\Vert {\overline{h}}_N(j\cdot ){\Vert }_{L^1(0,1)}=\Vert h_N{\Vert }_{L^1}\le \Vert f{\Vert }_{L^1}$. This proves the estimate. In particular, if zero-mean $g_j$ satisfy $\Vert g_j{\Vert }_{L^1}=O(j^m)$, their periodic extensions satisfy ${\overline{g}}_j(j\cdot )\to 0$ in weak convergence of distributions by choosing $N=m+1$.

Target: Essential closure. Parent: Lebesgue measure. Intended file: measure-theory.bigb.