A Generalized Sampling Theory

In this chapter we propose a regular sampling theory for a multiply generated U-invariant subspace of a separable Hilbert space \(\mathcal {H}\) . By regular sampling we mean that the samples are taking following the pattern given by the action of a discrete abelian group G on \(\mathcal {H}\) by means of a unitary representation \(g\mapsto U(g)\) of the group G on \(\mathcal {H}\) . This sampling process is carried out by following a suitable convolution system. Some frame properties related to this convolution system allow to derive a general sampling result. Thus, various different sampling settings can be considered as particular examples illustrating the sampling results in this chapter.