  We consider the problem of recovering a linear combination of Dirac delta
functions and derivatives from a finite number of Fourier samples corrupted by
noise. This is a generalized version of the well-known spike recovery problem,
which is receiving much attention recently.
  We analyze the numerical conditioning of this problem in two different
settings depending on the order of magnitude of the quantity $N\eta$, where $N$
is the number of Fourier samples and $\eta$ is the minimal distance between the
generalized spikes. In the "well-conditioned" regime $N\eta\gg1$, we provide
upper bounds for first-order perturbation of the solution to the corresponding
least-squares problem. In the near-colliding, or "super-resolution" regime
$N\eta\to0$ with a single cluster, we propose a natural regularization scheme
based on decimating the samples \textendash{} essentially increasing the
separation $\eta$ \textendash{} and demonstrate the effectiveness and
near-optimality of this scheme in practice.
