Generalized inverses and sampling problems

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