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We introduce a generalized framework for sampling and reconstruction inseparable Hilbert spaces. Specifically, we establish that it is always possibleto stably reconstruct a vector in an arbitrary Riesz basis from sufficientlymany of its samples in any other Riesz basis. This framework can be viewed asan extension of that of Eldar et al. However, whilst the latter imposesstringent assumptions on the reconstruction basis, and may in practice beunstable, our framework allows for recovery in any (Riesz) basis in a mannerthat is completely stable.  Whilst the classical Shannon Sampling Theorem is a special case of ourtheorem, this framework allows us to exploit additional information about theapproximated vector (or, in this case, function), for example sparsity orregularity, to design a reconstruction basis that is better suited. Examplesare presented illustrating this procedure.

A Generalized Sampling Theorem for Stable Reconstructions in Arbitrary Bases