Smooth-window Fourier coefficients of a periodic distribution ($c_m=\widehat{\chi f}(2\pi m)$)

A smooth compactly supported window with ${\sum }_{j\in \mathbb{Z}}\chi (x+j)=1$ permits multiplying every periodic distribution unambiguously. Its Fourier transform sampled at integer harmonics gives the periodic coefficients. The sharp window $H(x)-H(x-1)$ works for locally integrable functions but can be undefined at singular period endpoints, such as delta derivatives.