  Compressive sensing (CS) has recently emerged as a framework for efficiently
capturing signals that are sparse or compressible in an appropriate basis.
While often motivated as an alternative to Nyquist-rate sampling, there remains
a gap between the discrete, finite-dimensional CS framework and the problem of
acquiring a continuous-time signal. In this paper, we attempt to bridge this
gap by exploiting the Discrete Prolate Spheroidal Sequences (DPSS's), a
collection of functions that trace back to the seminal work by Slepian, Landau,
and Pollack on the effects of time-limiting and bandlimiting operations. DPSS's
form a highly efficient basis for sampled bandlimited functions; by modulating
and merging DPSS bases, we obtain a dictionary that offers high-quality sparse
approximations for most sampled multiband signals. This multiband modulated
DPSS dictionary can be readily incorporated into the CS framework. We provide
theoretical guarantees and practical insight into the use of this dictionary
for recovery of sampled multiband signals from compressive measurements.
