A new computational approach is presented, which allows fast analysis of radiation from reflector antennas. The basic idea is the Fourier transform (FT) relationship between far-field and aperture distributions. Accordingly, the far field can be exactly reconstructed from the knowledge of spatial samples at angular rate ka sin θ cosϕ = n π ka sin θ sin ϕ = m π, where a is the aperture radius (Shannon-Whittaker theorem applied at Nyquist rate). The conclusion is that the aperture far field can be sampled at approximately one point per lobe. The finite reflector curvature introduces an extra term in the radiation integral which modifies the (finite) aperture/far-field FT relationship. There are two ways to overcome this difficulty: (1) consider an infinite aperture, thereby saving the FT relationship, but losing the spatially band-limited property and (2) reformulate the Shannon-Whittaker theorem in such a way as to take into account the extra (curvature) term. The first approach requires truncation of the infinite aperture to an assigned degree of approximation in the far-field computation. This states an equivalence between the field radiated by the reflector and the field radiated by an equivalent aperture of larger radius. Accordingly, the far field can still be reconstructed using the field radiated by the reflector, sampled at a higher rate (slightly higher for large dishes). These samples can be computed using physical optics or asymptotic techniques, whichever is appropriate. The second approach is based on a new sampling representation, which explicitly takes into account the extra term in the radiation integral. For a quadratic type of term, i.e., for parabolic reflectors, the sampling functions are the complementary error functions, and the far field can be exactly reconstructed in terms of the far field of an equivalent aperture. These samples can be computed using the fast Fourier transform, therefore further gaining in time saving. Computations showing the excellent performances of the two methods have been made, indicating the significant advantages provided in reflector antenna analysis.