  We study the problem of recovering an unknown compactly-supported
multivariate function from samples of its Fourier transform that are acquired
nonuniformly, i.e. not necessarily on a uniform Cartesian grid. Reconstruction
problems of this kind arise in various imaging applications, where Fourier
samples are taken along radial lines or spirals for example.
  Specifically, we consider finite-dimensional reconstructions, where a limited
number of samples is available, and investigate the rate of convergence of such
approximate solutions and their numerical stability. We show that the
proportion of Fourier samples that allow for stable approximations of a given
numerical accuracy is independent of the specific sampling geometry and is
therefore universal for different sampling scenarios. This allows us to relate
both sufficient and necessary conditions for different sampling setups and to
exploit several results that were previously available only for very specific
sampling geometries.
  The results are obtained by developing: (i) a transference argument for
different measures of the concentration of the Fourier transform and Fourier
samples; (ii) frame bounds valid up to the critical sampling density, which
depend explicitly on the sampling set and the spectrum.
  As an application, we identify sufficient and necessary conditions for stable
and accurate reconstruction of algebraic polynomials or wavelet coefficients
from nonuniform Fourier data.
