On stable reconstructions from nonuniform Fourier measurements

TL;DR

This paper investigates stable methods for reconstructing functions from nonuniform Fourier samples, establishing conditions for accuracy and demonstrating wavelets as optimal spaces for such reconstructions.

Contribution

It provides theoretical conditions for stable reconstruction from nonuniform Fourier data and proves wavelets are optimal approximation spaces in this context.

Findings

Stable reconstruction is possible under certain sampling density conditions.

Wavelets require linear scaling of dimension with bandwidth for stable recovery.

Wavelets are proven to be optimal approximation spaces for this problem.

Abstract

We consider the problem of recovering a compactly-supported function from a finite collection of pointwise samples of its Fourier transform taking nonuniformly. First, we show that under suitable conditions on the sampling frequencies - specifically, their density and bandwidth - it is possible to recover any such function $f$ in a stable and accurate manner in any given finite-dimensional subspace; in particular, one which is well suited for approximating $f$. In practice, this is carried out using so-called nonuniform generalized sampling (NUGS). Second, we consider approximation spaces in one dimension consisting of compactly supported wavelets. We prove that a linear scaling of the dimension of the space with the sampling bandwidth is both necessary and sufficient for stable and accurate recovery. Thus wavelets are up to constant factors optimal spaces for reconstruction.