Beyond consistent reconstructions: optimality and sharp bounds for generalized sampling, and application to the uniform resampling problem

摘要

Generalized sampling is a recently developed linear framework for samplingand reconstruction in separable Hilbert spaces. It allows one to recover anyelement in any finite-dimensional subspace given finitely many of its sampleswith respect to an arbitrary frame. Unlike more common approaches for thisproblem, such as the consistent reconstruction technique of Eldar et al, itleads to completely stable numerical methods possessing both guaranteedstability and accuracy. The purpose of this paper is twofold. First, we give a complete and formalanalysis of generalized sampling, the main result of which being the derivationof new, sharp bounds for the accuracy and stability of this approach. Suchbounds improve those given previously, and result in a necessary and sufficientcondition, the stable sampling rate, which guarantees a priori a goodreconstruction. Second, we address the topic of optimality. Under someassumptions, we show that generalized sampling is an optimal, stablereconstruction. Correspondingly, whenever these assumptions hold, the stablesampling rate is a universal quantity. In the final part of the paper weillustrate our results by applying generalized sampling to the so-calleduniform resampling problem.