  Line spectral estimation is the problem of recovering the frequencies and
amplitudes of a mixture of a few sinusoids from equispaced samples. However, in
a variety of signal processing problems arising in imaging, radar, and
localization we do not have access directly to such equispaced samples. Rather
we only observe a severely undersampled version of these observations through
linear measurements. This paper is about such generalized line spectral
estimation problems. We reformulate these problems as sparse signal recovery
problems over a continuously indexed dictionary which can be solved via a
convex program. We prove that the frequencies and amplitudes of the components
of the mixture can be recovered perfectly from a near-minimal number of
observations via this convex program. This result holds provided the
frequencies are sufficiently separated, and the linear measurements obey
natural conditions that are satisfied in a variety of applications.
