  This paper presents generalizations of semidefinite programming formulations
of 1-norm optimization problems over infinite dictionaries of vectors of
complex exponentials, which were recently proposed for superresolution,
gridless compressed sensing, and other applications in signal processing.
Results related to the generalized Kalman-Yakubovich-Popov lemma in linear
system theory provide simple, constructive proofs of the semidefinite
representations of the penalty functions used in these applications. The
connection leads to several extensions to gauge functions and atomic norms for
sets of vectors parameterized via the nullspace of matrix pencils. The
techniques are illustrated with examples of low-rank matrix approximation
problems arising in spectral estimation and array processing.
