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

TL;DR

This paper provides a comprehensive analysis of generalized sampling, deriving sharp bounds for its stability and accuracy, establishing its optimality under certain conditions, and demonstrating its application to the uniform resampling problem.

Contribution

It offers a complete formal analysis of generalized sampling, introduces sharp bounds for stability and accuracy, and proves its optimality under specific assumptions.

Findings

Derived new, sharp bounds for accuracy and stability of generalized sampling.

Established necessary and sufficient conditions for stable reconstruction.

Applied generalized sampling to the uniform resampling problem.

Abstract

Generalized sampling is a recently developed linear framework for sampling and reconstruction in separable Hilbert spaces. It allows one to recover any element in any finite-dimensional subspace given finitely many of its samples with respect to an arbitrary frame. Unlike more common approaches for this problem, such as the consistent reconstruction technique of Eldar et al, it leads to completely stable numerical methods possessing both guaranteed stability and accuracy. The purpose of this paper is twofold. First, we give a complete and formal analysis of generalized sampling, the main result of which being the derivation of new, sharp bounds for the accuracy and stability of this approach. Such bounds improve those given previously, and result in a necessary and sufficient condition, the stable sampling rate, which guarantees a priori a good reconstruction. Second, we address the topic of optimality. Under some assumptions, we show that generalized sampling is an optimal, stable reconstruction. Correspondingly, whenever these assumptions hold, the stable sampling rate is a universal quantity. In the final part of the paper we illustrate our results by applying generalized sampling to the so-called uniform resampling problem.