My advisor recently asked me a series of questions that made me realize I don't fully understand how the fourier transform of, lets say series of vertical lines, produces a diffraction pattern like those seen in a TEM. To get a better understanding I re-read Hecht's short little chapter on fourier series, integrals, and their corresponding graphs in k-space. I understand that Fourier series allow us to write any equation in terms of a series of sines and cosines, and that calculating the related coefficients tells us how each frequency of each of the sines and cosines is weighted into the representation of the equation. I'm not, however, seeing how taking the fourier series, fourier transform, or diffraction patterns are related. Could anyone help with the missing link?