Exact Iterative Reconstruction Algorithm for Multivariate Irregularly Sampled Functions in Spline-like Spaces: the L-theory

We prove that the exact reconstruction of a function s from its samples s(xi) on any “sufficiently dense” sampling set {xi}i∈Λ can be obtained, as long as s is known to belong to a large class of spline-like spaces in Lp(Rn). Moreover, the reconstruction can be implemented using fast algorithms. Since a limiting case is the space of bandlimited functions, our result generalizes the classical Shannon-Whittaker sampling theorem on regular sampling and the Paley-Wiener theorem on non-uniform sampling.