Super-resolution of point sources via convex programming

We consider the problem of recovering a signal consisting of a superposition of point sources from low-resolution data with a cutoff frequency f(c). If the distance between the sources is under 1/f(c), 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 l(1)-norm achieves exact recovery as long as the sources are separated by at least 1.26/f(c). 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.