Smooth-window Fourier coefficients of a periodic distribution
= Smooth-window Fourier coefficients of a periodic distribution
{title2=$c_m=\widehat{\chi f}(2\pi m)$}
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.