= Periodic primitive with zero mean
{title2=$Ph(x)=-\int_0^xh(s)\,ds+\int_0^1\int_0^yh(s)\,ds\,dy$}
= Periodic primitives with zero mean
{synonym}
For $h\in L^1(0,1)$ with integral zero, $Ph$ is the unique <absolutely continuous function> with $(Ph)'=-h$, equal endpoint values, and integral zero. The first formula gives the <derivative>; zero integral of $h$ gives equal endpoint values, while the added constant makes the mean zero. Uniqueness follows because the difference of two such primitives has zero <weak derivative> and therefore is constant, then its mean is zero. Its <periodic extension> has <distributional derivative> $-\overline h$, with no endpoint <Dirac delta distribution>.
Writing $Rh(x)=x\int_0^1h-\int_0^xh$, the same operator on zero-mean inputs is $Ph=Rh-\int_0^1Rh$. For arbitrary $h\in L^1(0,1)$,
$$
\|Rh\|_{L^1}\leq\frac12\|h\|_{L^1},\qquad
\left\|Rh-\int_0^1Rh\right\|_{L^1}\leq\|h\|_{L^1}.
$$
The first inequality follows from the integral kernel $x-\mathbf1_{\{t<x\}}$, whose absolute value has integral $(t^2+(1-t)^2)/2\leq1/2$ in $x$. The second follows by the <triangle inequality>. This bound permits arbitrarily many iterations without increasing the <L1 space> norm.
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Target: Rapid distributional decay of periodic oscillations. Parent: Weak convergence of distributions. Intended file: distribution-theory.bigb.
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