Optimal learning of bandlimited functions from localized sampling

An optimal algorithm for approximating bandlimited functions from localized sampling is established. Several equivalent formulations for the approximation error of the optimal algorithm are presented and its upper and lower bound estimates for the univariate case are provided. The estimates show that the approximation error decays exponentially (but not faster) as the number of localized samplings increases. As a consequence of these results, we obtain an upper bound estimate for the eigenvalues of an integral operator that arises in the bandwidth problem.