  We introduce a generalized framework for sampling and reconstruction in
separable Hilbert spaces. Specifically, we establish that it is always possible
to stably reconstruct a vector in an arbitrary Riesz basis from sufficiently
many of its samples in any other Riesz basis. This framework can be viewed as
an extension of that of Eldar et al. However, whilst the latter imposes
stringent assumptions on the reconstruction basis, and may in practice be
unstable, our framework allows for recovery in any (Riesz) basis in a manner
that is completely stable.
  Whilst the classical Shannon Sampling Theorem is a special case of our
theorem, this framework allows us to exploit additional information about the
approximated vector (or, in this case, function), for example sparsity or
regularity, to design a reconstruction basis that is better suited. Examples
are presented illustrating this procedure.
