Smooth periodic partition of unity (${\sum }_{k\in \mathbb{Z}}\Psi (x+kT)=1$)

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.