# Sampling and Reconstruction in Distinct Subspaces Using Oblique Projections

## TL;DR

This paper investigates optimal reconstruction operators in Hilbert spaces that balance stability and accuracy, with applications to nonuniform Fourier sampling, by analyzing oblique projections and their properties.

## Contribution

It introduces methods to construct and interpolate between reconstruction operators optimized for stability and quasi-optimality in subspace sampling.

## Key findings

- Reconstruction operator with minimal operator norm is most stable.
- Operator with minimal quasi-optimality constant offers robustness against systematic errors.
- Continuous variation between operators allows trading stability for accuracy.

## Abstract

We study reconstruction operators on a Hilbert space that are exact on a given reconstruction subspace. Among those the reconstruction operator obtained by the least squares fit has the smallest operator norm, and therefore is most stable with respect to noisy measurements. We then construct the operator with the smallest possible quasi-optimality constant, which is the most stable with respect to a systematic error appearing before the sampling process (model uncertainty). We describe how to vary continuously between the two reconstruction methods, so that we can trade stability for quasi-optimality. As an application we study the reconstruction of a compactly supported function from nonuniform samples of its Fourier transform.