  We provide sufficient density condition for a set of nonuniform samples to
give rise to a set of sampling for multivariate bandlimited functions when the
measurements consist of pointwise evaluations of a function and its first $k$
derivatives. Along with explicit estimates of corresponding frame bounds, we
derive the explicit density bound and show that, as $k$ increases, it grows
linearly in $k+1$ with the constant of proportionality $1/\mathrm{e}$. Seeking
larger gap conditions, we also prove a multivariate perturbation result for
nonuniform samples that are sufficiently close to sets of sampling, e.g. to
uniform samples taken at $k+1$ times the Nyquist rate.
  Additionally, in the univariate setting, we consider a related problem of
so-called nonuniform bunched sampling, where in each sampling interval $s+1$
bunched measurements of a function are taken and the sampling intervals are
permitted to be of different length. We derive an explicit density condition
which grows linearly in $s+1$ for large $s$, with the constant of
proportionality depending on the width of the bunches. The width of the bunches
is allowed to be arbitrarily small, and moreover, for sufficiently narrow
bunches and sufficiently large $s$, we obtain the same result as in the case of
univariate sampling with $s$ derivatives.
