  This paper presents new theoretical results on sparse recovery guarantees for
a greedy algorithm, Orthogonal Matching Pursuit (OMP), in the context of
continuous parametric dictionaries. Here, the continuous setting means that the
dictionary is made up of an infinite uncountable number of atoms. In this work,
we rely on the Hilbert structure of the observation space to express our
recovery results as a property of the kernel defined by the inner product
between two atoms. Using a continuous extension of Tropp's Exact Recovery
Condition, we identify key assumptions allowing to analyze OMP in the
continuous setting. Under these assumptions, OMP unambiguously identifies in
exactly $k$ steps the atom parameters from any observed linear combination of
$k$ atoms. These parameters play the role of the so-called support of a sparse
representation in traditional sparse recovery. In our paper, any kernel and set
of parameters that satisfy these conditions are said to be admissible. In the
one-dimensional setting, we exhibit a family of kernels relying on completely
monotone functions for which admissibility holds for any set of atom
parameters. For higher dimensional parameter spaces, the analysis turns out to
be more subtle. An additional assumption, so-called axis admissibility, is
imposed to ensure a form of delayed recovery (in at most $k^D$ steps, where $D$
is the dimension of the parameter space). Furthermore, guarantees for recovery
in exactly $k$ steps are derived under an additional algebraic condition
involving a finite subset of atoms (built as an extension of the set of atoms
to be recovered). We show that the latter technical conditions simplify in the
case of Laplacian kernels, allowing us to derive simple conditions for $k$-step
exact recovery, and to carry out a coherence-based analysis in terms of a
minimum separation assumption between the atoms to be recovered.
