  In single-molecule microscopy it is necessary to locate with high precision
point sources from noisy observations of the spectrum of the signal at
frequencies capped by $f_c$, which is just about the frequency of natural
light. This paper rigorously establishes that this super-resolution problem can
be solved via linear programming in a stable manner. We prove that the quality
of the reconstruction crucially depends on the Rayleigh regularity of the
support of the signal; that is, on the maximum number of sources that can occur
within a square of side length about $1/f_c$. The theoretical performance
guarantee is complemented with a converse result showing that our simple convex
program convex is nearly optimal. Finally, numerical experiments illustrate our
methods.
