We discuss extensions of recently proposed semidefinite optimization methods for atomic decomposition and 1-norm minimization over infinite dictionaries of complex exponentials. We show that techniques related to the Kalman-Yakubovich-Popov lemma in linear system theory provide simple constructive proofs of these semidefinite formulations. This connection directly leads to extensions to more general dictionaries associated with state-space models and matrix pencils. The results will be illustrated with applications in signal processing.