  The problem of super-resolution in general terms is to recuperate a finitely
supported measure $\mu$ given finitely many of its coefficients $\hat{\mu}(k)$
with respect to some orthonormal system. The interesting case concerns
situations, where the number of coefficients required is substantially smaller
than a power of the reciprocal of the minimal separation among the points in
the support of $\mu$. In this paper, we consider the more severe problem of
recuperating $\mu$ approximately without any assumption on $\mu$ beyond having
a finite total variation. In particular, $\mu$ may be supported on a continuum,
so that the minimal separation among the points in the support of $\mu$ is $0$.
A variant of this problem is also of interest in machine learning as well as
the inverse problem of de-convolution. We define an appropriate notion of a
distance between the target measure and its recuperated version, give an
explicit expression for the recuperation operator, and estimate the distance
between $\mu$ and its approximation. We show that these estimates are the best
possible in many different ways. We also explain why for a finitely supported
measure the approximation quality of its recuperation is bounded from below if
the amount of information is smaller than what is demanded in the
super-resolution problem.
