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 s from its samples s ( x i ) s (x_i) on any “sufficiently dense" sampling set { x i } i ∈ Λ \{x_i\}i∈ Λ can be obtained, as long as s s is known to belong to a large class of spline-like spaces in L p ( R n ) 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.