Numerical simulations of charged particle beams require an approximation to the particle distribution being simulated. Depending on the flavor of the N-body code, these approximations suffer from different computational difficulties. In this talk, we briefly outline these difficulties, and present approximations to particle distributions using orthogonal functions. We discuss two different types of orthogonal functions, new in the context of beam simulations: wavelets and scaled Gauss-Hermite basis. On the wavelet side, we present the wavelet-based Poisson equation solver we recently devised for use in particle-in-cell beam simulations, and report on some important enhancements being implemented as a part of an ongoing project. On scaled Gauss-Hermite basis side, we discuss some preliminary results in efficiently approximating discrete particle distributions in an orthogonal basis in which the corresponding potential and forces are directly and easily found from the expansion coefficients of the distribution.