Why does direct-MUSIC on sparse-arrays work?

The nested and coprime arrays have recently been introduced as systematic structures to construct difference coar-rays with O(m^2) elements, where m is the number of array elements. They are therefore able to identify O(m^2) sources (or DOAs) under the assumption that the sources are uncorrelated. In view of their larger aperture compared to uniform linear arrays (ULAs) with the same number of elements, these arrays have some advantages over conventional ULAs even in the cases where the number of sources is less than m, such as improved Cramer-Rao bounds and improved resolvability for closely spaced sources. It has recently been shown that in such situations and under mild assumptions on source locations, it is possible to use subspace techniques such as the MUSIC algorithm directly on these sparse arrays, instead of on the coarrays, thereby making the resulting algorithms simpler. Thus the ambiguity introduced by the sparsity of the arrays is overcome with the help of extra elements with coprime spacing, even if there are multiple sources. The purpose of this paper is to give the theoretical justification for this