Dirac comb (${\sum }_{j\in \mathbb{Z}}\delta (x-jD)$)

A Dirac comb is a periodic train of Dirac delta distributions. With Fourier transform convention $\hat{f}(q)=\int f(x)e^{-iqx}dx$ and period $D>0$,The identity holds as a tempered distribution. Truncating the real-space train by a finite window convolves the reciprocal train with the transform of that window, broadening its peaks. Convolving the train with one slit's transmission instead multiplies the reciprocal peaks by the slit's field transform, the basic construction of a diffraction grating pattern.