  We consider the problem of recovering a signal consisting of a superposition
of point sources from low-resolution data with a cut-off frequency f. If the
distance between the sources is under 1/f, this problem is not well posed in
the sense that the low-pass data corresponding to two different signals may be
practically the same. We show that minimizing a continuous version of the l1
norm achieves exact recovery as long as the sources are separated by at least
1.26/f. The proof is based on the construction of a dual certificate for the
optimization problem, which can be used to establish that the procedure is
stable to noise. Finally, we illustrate the flexibility of our
optimization-based framework by describing extensions to the demixing of sines
and spikes and to the estimation of point sources that share a common support.
