In single-molecule microscopy it is necessary to reconstruct a signal that consists of positive point sources from noisy observations of the spectrum of the signal in the low-frequency band [−fc,fc]. It is shown that the problem can be solved using convex optimization in a stable fashion. The stability of reconstruction depends on Rayleigh-regularity of the support of the signal, i.e., on how many point sources can occur within an interval of length 1.87/fc. The stability estimate is complimented by a converse result: the performance of convex algorithm is nearly optimal. The results are generalized to multi-dimension signals. Applications in microscopy are briefly discussed.