On the Role of the Bounded Lemma in the SDP Formulation of Atomic Norm Problems

In problems involving the optimization of atomic norms, an upper bound on the dual atomic norm often arises as a constraint. For the special case of line spectral estimation, this upper bound on the dual atomic norm reduces to upper-bounding the magnitude response of a finite impulse response filter by a constant. It is well known that this can be rewritten as a semidefinite constraint, leading to an elegant semidefinite programming formulation of the atomic norm minimization problem. This result is a direct consequence of some classical results in system theory, well known for many decades. This is not detailed in the literature on atomic norms, quite understandably, because the emphasis therein is different. In fact, these connections can be found in the book by B. A. Dumitrescu, cited widely in the atomic norm literature. However, they are spread out among many different results and formulations. This letter makes the connection more clear by appealing to one simple result from system theory, thereby making it more transparent to wider audience.