Sampling theory, oblique projections and a question by Smale and Zhou

Abstract

In a recent article, Smale and Zhou define a notion of rich data for sampling problems and reconstruction of signals from a discrete set of samples and study different least-square problems related with the minimization of the error. They obtain different error estimations assuming that the original signal belong to the reconstruction subspace and they propose to find error estimations if this assumption does not hold. In this paper, using projection methods, we find such estimates and we extend from reproducing kernel Hilbert spaces to abstract Hilbert spaces some of their results on function reconstruction from point values.