This paper describes a dual certificate condition on a linear measurement operator A (defined on a Hilbert space H and having finite-dimensional range) which guarantees that an atomic norm minimization, in a certain sense, will be able to approximately recover a structured signal v0∈H from measurements Av0. Put very streamlined, the condition implies that peaks in a sparse decomposition of v0 are close the the support of the atomic decomposition of the solution v∗. The condition applies in a relatively general context - in particular, the space H can be infinite-dimensional. The abstract framework is applied to several concrete examples, one example being super-resolution. In this process, several novel results which are interesting on its own are obtained.
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