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

We prove that the exact reconstruction of a function from its samples on any ``sufficiently dense" sampling set can be obtained, as long as is known to belong to a large class of spline-like spaces in . 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.