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If $f\in L^1(0,1)$ has mean zero, $j$ is a positive integer and $\varphi \in \mathcal{D}(0,1)$, then for every integer $N\ge 1$,Here the overline denotes the periodic extension, not complex conjugation. Let $h_N=P^{N}f$ use periodic primitives with zero mean. Repeated integration by parts gives the signed identityThere are no boundary terms, and integration over exactly $j$ periods gives $\Vert {\overline{h}}_N(j\cdot ){\Vert }_1=\Vert h_N{\Vert }_1\le \Vert f{\Vert }_1$. This proves the estimate. If zero-mean $g_j$ satisfy $\Vert g_j{\Vert }_1=O(j^m)$ for fixed real $m\ge 0$, choose an integer $N>m$ to obtain ${\overline{g}}_j(j\cdot )\to 0$ in weak convergence of distributions.

A triangular wave is a continuous piecewise-linear periodic function with successive rising and falling segments. For $b>0$, the even $2\pi$-periodic extension of $bx$ from $[0,\pi ]$ isIt has troughs of height zero at $2k\pi$, peaks of height $b\pi$ at $(2k+1)\pi$, and slopes $-b$ and $b$ between the corners. The function is continuous at the corners but lacks a derivative there.

Its Fourier cosine series contains a constant term and odd modes with coefficients proportional to the inverse square of the mode number. The Fourier cosine series of a triangular wave computes this for another normalization. Even periodic extensions also encode reflection in problems with Neumann boundary conditions.