Infinite-dimensional compressed sensing and function interpolation

We introduce and analyze a framework for function interpolation using compressed sensing. This framework – which is based on weighted l minimization – does not require a priori bounds on the expansion tail in either its implementation or its theoretical guarantees. Moreover, in the absence of noise it leads to genuinely interpolatory approximations. We also establish a series of new recovery guarantees for compressed sensing with weighted l minimization based on this framework. These guarantees convey three key benefits. First, unlike some previous results, they are sharp (up to constants and log factors) for large classes of functions regardless of the choice of weights. Second, they allow one to determine a provably optimal weighting strategy in the case of multivariate polynomial approximations in lower sets. Third, they can be used to establish the benefits of weighting strategies where the weights are chosen based on prior support information, thus providing a theoretical basis for a number of numerical studies which have shown this to be the case.