  Generalized sampling is a recently developed linear framework for sampling
and reconstruction in separable Hilbert spaces. It allows one to recover any
element in any finite-dimensional subspace given finitely many of its samples
with respect to an arbitrary frame. Unlike more common approaches for this
problem, such as the consistent reconstruction technique of Eldar et al, it
leads to completely stable numerical methods possessing both guaranteed
stability and accuracy.
  The purpose of this paper is twofold. First, we give a complete and formal
analysis of generalized sampling, the main result of which being the derivation
of new, sharp bounds for the accuracy and stability of this approach. Such
bounds improve those given previously, and result in a necessary and sufficient
condition, the stable sampling rate, which guarantees a priori a good
reconstruction. Second, we address the topic of optimality. Under some
assumptions, we show that generalized sampling is an optimal, stable
reconstruction. Correspondingly, whenever these assumptions hold, the stable
sampling rate is a universal quantity. In the final part of the paper we
illustrate our results by applying generalized sampling to the so-called
uniform resampling problem.
