  In this paper, we study the spectral estimation problem of estimating the
locations of a fixed number of point sources given multiple snapshots of
Fourier measurements in a bounded domain. We aim to provide a mathematical
foundation for sparsity-based super-resolution in such spectral estimation
problems in both one- and multi-dimensional spaces. In particular, we estimate
the resolution and stability of the location recovery when considering the
sparsest solution under the measurement constraint, and characterize their
dependence on the cut-off frequency, the noise level, the sparsity of point
sources, and the incoherence of the amplitude vectors of point sources. Our
estimate emphasizes the importance of the high incoherence of amplitude vectors
in enhancing the resolution of multi-snapshot spectral estimation. Moreover, to
the best of our knowledge, it also provides the first stability result in the
super-resolution regime for the well-known sparse MMV problem in DOA
estimation.
