Periodic primitive with zero mean ($Ph(x)=-{\int }_0^{x}h(s)ds+{\int }_0^1{\int }_0^{y}h(s)dsdy$)

For $h\in L^1(0,1)$ with integral zero, $Ph$ is the unique absolutely continuous function with $(Ph{)}^{\prime }=-h$, 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 $Rh(x)=x{\int }_0^{1}h-{\int }_0^{x}h$, the same operator on zero-mean inputs is $Ph=Rh-{\int }_0^{1}Rh$. For arbitrary $h\in L^1(0,1)$,The first inequality follows from the integral kernel $x-1_{\{t<x\}}$, whose absolute value has integral $(t^2+(1-t{)}^2)/2\le 1/2$ in $x$. The second follows by the triangle inequality. This bound permits arbitrarily many iterations without increasing the L1 space norm.

Target: Rapid distributional decay of periodic oscillations. Parent: Weak convergence of distributions. Intended file: distribution-theory.bigb.