Generalized Sampling: A Variational Approach

We consider the problem of reconstructing a multidimensional and multivariate function f : R → R from the discretely and irregularly sampled responses of q linear shift-invariant filters. Unlike traditional approaches which reconstruct the function in some signal space V , our reconstruction is optimal in the sense of a plausibility criterion J . The reconstruction is either consistent with the measures, or minimizes the consistence error. There is no band-limiting restriction for the input signals. We show that important characteristics of the reconstruction process are induced by the properties of the criterion J . We give the reconstruction formula and apply it to several practical cases.