Sampling in Shift-Invariant Subspaces

For avoiding the drawbacks that have the bandlimited functions, it was proposed sampling and reconstruction problems in shift-invariant subspaces of \(L^2(\mathbb {R})\) functions. Once the basic properties of these spaces has been studied, Shannon-type sampling formulas are derived. These formulas are, in general, Riesz bases expansions. Under an appropriate manipulation of the samples which uses differences or averages, any Shannon-type formula can be expressed in terms of the new sampling data. Finally, we consider the generalization of shift-invariant spaces that consists of replacing the shift operator by another unitary operator U on a separable Hilbert space obtaining the U-sampling concept.