  In this paper, we provide a Rapid Orthogonal Approximate Slepian Transform
(ROAST) for the discrete vector that one obtains when collecting a finite set
of uniform samples from a baseband analog signal. The ROAST offers an
orthogonal projection which is an approximation to the orthogonal projection
onto the leading discrete prolate spheroidal sequence (DPSS) vectors (also
known as Slepian basis vectors). As such, the ROAST is guaranteed to accurately
and compactly represent not only oversampled bandlimited signals but also the
leading DPSS vectors themselves. Moreover, the subspace angle between the ROAST
subspace and the corresponding DPSS subspace can be made arbitrarily small. The
complexity of computing the representation of a signal using the ROAST is
comparable to the FFT, which is much less than the complexity of using the DPSS
basis vectors. We also give non-asymptotic results to guarantee that the
proposed basis not only provides a very high degree of approximation accuracy
in a mean squared error sense for bandlimited sample vectors, but also that it
can provide high-quality approximations of all sampled sinusoids within the
band of interest.
