In this paper, we apply signal approximation theory to estimate the articulatory trajectory for a nonlinear speech synthesizer. First, we analyze the L2 error bounds of interpolation and LS approximation in the Hilbert space framework. Second, we use two signal approximation techniques, specifically, cubic spline interpolation and the LS approximation to estimate the trajectory of articulatory parameters given the static articulatory parameters of discrete phonemes. In our articulatory speech synthesizer, we assume that the sound propagation inside the human vocal tract is a three-dimensional non-plane wave propagation inside a viscous uid described by the governing Navier-Stoke equations. Finally, we present some experimental results of the estimated articulatory trajectories of English diphthongs and the synthesized phonemes.