Spectral decomposition of grid data

Spectral decomposition is a method of expressing functions as a harmonic series, and can be used for the simplification of complicated physical problems. This type of analysis requires knowledge of the function at all points on a circle or sphere. In problems where the function is known only at discreet points, regular intervals in a rectangular grid, for example, numerical methods must be employed to compute approximate coefficients for the harmonic expansion. In this paper, we investigate numerical methods for computing Fourier coefficients of a two dimensional function at a fixed radius, and spherical harmonic coefficients in three dimensions on a sphere of fixed radius.