A Generalized Sampling Theorem for Stable Reconstructions in Arbitrary Bases

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.