= Smooth periodic partition of unity
{title2=$\sum_{k\in\mathbb Z}\Psi(x+kT)=1$}
= Compactly supported periodic window
{synonym}
For each $T>0$ there is a nonnegative <test function> $\Psi$ with the displayed identity. Choose a nonnegative <bump function> $\eta$ supported in $[-T,T]$ and positive on $[-T/2,T/2]$. The sum $S(x)=\sum_k\eta(x+kT)$ is a positive, <smooth>, $T$ <periodic function>. Setting $\Psi=\eta/S$ preserves <compact support> and smoothness, and periodicity of $S$ proves $\sum_k\Psi(x+kT)=S(x)/S(x)=1$. The translates form a locally finite <partition of unity> on the real line.
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