  We consider the problem of reconstructing one-dimensional point sources from
their Fourier measurements in a bounded interval $[-\Omega, \Omega]$. This
problem is known to be challenging in the regime where the spacing of the
sources is below the Rayleigh length $\frac{\pi}{\Omega}$. In this paper, we
propose a super-resolution algorithm, called Iterative Focusing-localization
and Filtering (IFF), to resolve closely spaced point sources from their
multiple measurements that are obtained by using multiple unknown illumination
patterns. The new proposed algorithm has a distinct feature in that it
reconstructs the point sources one by one in an iterative manner and hence
requires no prior information about the source numbers. The new feature also
allows for a subsampling strategy that can circumvent the computation of
singular-value decomposition for large matrices as in the usual subspace
methods. A theoretical analysis of the methods behind the algorithm is also
provided. The derived results imply a phase transition phenomenon in the
reconstruction of source locations which is confirmed in numerical experiments.
Numerical results show that the algorithm can achieve a stable reconstruction
for point sources with a minimum separation distance that is close to the
theoretical limit. The algorithm can be generalized to higher dimensions.
