The discrete prolate spheroidal sequences (DPSS's) provide an efficient representation for signals that are perfectly timelimited and nearly bandlimited. Unfortunately, because of the high computational complexity of projecting onto the DPSS basis -- also known as the Slepian basis -- this representation is often overlooked in favor of the fast Fourier transform (FFT). In this presentation, we show that there exist fast constructions for computing approximate projections onto the leading Slepian basis elements. The complexity of the resulting algorithms is comparable to the FFT, and scales favorably as the quality of the desired approximation is increased. We demonstrate how these algorithms allow us to efficiently compute the solution to certain least-squares problems that arise in signal processing.