Complete iterative reconstruction algorithms for irregularly sampled data in spline-like spaces

We prove that the exact reconstruction of a function s from its samples s(x_i) on any “sufficiently dense” sampling set {x_i}_i∈I⊂Rⁿ, where I is a countable indexing set, can be obtained for a large class of spline-like spaces that belong to LP(Rⁿ). Moreover, the reconstruction can be implemented using fast algorithms. Since, a special case is the space of bandlimited functions, our result generalizes the classical Shannon-Whittacker (1949) sampling theorem on regular sampling and the Paley-Wiener (1934) theorem on nonuniform sampling