Approximate spectral representations are developed for the memory kernel which characterizes self-diffusion. These spectral representations are based upon approximate eigenfunctions constructed via the Rayleigh variational principle. A heuristic model is developed first in an effort to provide physical insight into the nature of the approximations employed, and then a number of specific trial functions are examined. These trial functions include sums of identical one- and two-particle functions as well as linear combinations of hydrodynamical variables. The results from these spectral representations indicate that the long-time behavior of the memory kernel (and thereby of the momentum autocorrelation function) is sensitive to the long-range effects of the interparticle potential. In addition, the equivalence of most of these spectral representations to specific low-order perturbation approximations is demonstrated.