  This paper describes a dual certificate condition on a linear measurement
operator $A$ (defined on a Hilbert space $\mathcal{H}$ 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 $v_0
\in \mathcal{H}$ from measurements $Av_0$. Put very streamlined, the condition
implies that peaks in a sparse decomposition of $v_0$ 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 $\mathcal{H}$ 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.
