Exact iterative reconstruction algorithm for multivariate irregularly sampled functions in spline-like spaces:: The Lp-theory

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 epsilon Lambda) can be obtained, as long as s is known to belong to a large class of spline-like spaces in L-p(R-n). 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.