FT and FFT

I was thinking of using a spectral method to simplify our CF solver. Since the Fourier coefficients of the dielectric function on a ring of a micro-cavity (e.g. quadruple) is just a step function, the integral can be solved analytically (which means FAST!). But when reconstructing the dielectric function, the result is not as good as I expected: it has fast oscillations of small amplitude, which grows towards the edge of the step function. It doesn't improve much even when I increase M to be 1000. On the other hand, FFT seems to do much better job even with N=200. The difference between the original function and the reconstructed one differ by only 10^-15! But I need to think of a way to deal with the FFT of a product (n^2*\phi). If the angular momentum is cut off at M for \phi, the FT of the product needs the Fourier coefficients of (n^2) upto be 2*M. How to translate that into the FFT? Does it mean the spatial grid points for (n^2) needs to be twice as many of the CF state \phi? I will keep you updated in later posts. Now it is soccer time!