  This paper is concerned about sparse, continuous frequency estimation in line
spectral estimation, and focused on developing gridless sparse methods which
overcome grid mismatches and correspond to limiting scenarios of existing
grid-based approaches, e.g., $\ell_1$ optimization and SPICE, with an
infinitely dense grid. We generalize AST (atomic-norm soft thresholding) to the
case of nonconsecutively sampled data (incomplete data) inspired by recent
atomic norm based techniques. We present a gridless version of SPICE (gridless
SPICE, or GLS), which is applicable to both complete and incomplete data
without the knowledge of noise level. We further prove the equivalence between
GLS and atomic norm-based techniques under different assumptions of noise.
Moreover, we extend GLS to a systematic framework consisting of model order
selection and robust frequency estimation, and present feasible algorithms for
AST and GLS. Numerical simulations are provided to validate our theoretical
analysis and demonstrate performance of our methods compared to existing ones.
