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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/sub i/) on any "sufficiently dense" sampling set {x/sub i/}/sub i/spl isin/I//spl sub/R/sup n/, where I is a countable indexing set, can be obtained for a large class of spline-like spaces that belong to LP(R/sup n/). 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.