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Sampling theory is concerned with the problem of reconstructing a signal f in a Hilbertspace from a given a collection of sampled values of f . If a certain decomposition of theHilbert space is possible (in terms of the sampling and reconstruction subspaces) thena consistent reconstruction can be obtained. In this paper we treat the case in whichsuch a decomposition cannot be found. For this situation, we study the quasi-consistentreconstructions which are an extension of the consistent reconstructions. We relate theprevious concepts to generalized inverses.We also present some new results and problemsregarding consistent sampling.