Recovering piecewise smooth functions from nonuniform Fourier measurements

TL;DR

This paper presents a method for accurately reconstructing piecewise smooth functions from nonuniform Fourier measurements using spline and polynomial-based reconstruction spaces within the NUGS framework.

Contribution

It analyzes the relationship between the reconstruction space dimension and sample bandwidth, providing new insights into optimal sampling strategies.

Findings

Reconstruction accuracy improves with appropriately chosen spline or polynomial spaces.

The dimension of the reconstruction space scales linearly with bandwidth for fixed-degree splines and polynomials.

The dimension scales quadratically for variable-degree piecewise polynomials.

Abstract

In this paper, we consider the problem of reconstructing piecewise smooth functions to high accuracy from nonuniform samples of their Fourier transform. We use the framework of nonuniform generalized sampling (NUGS) to do this, and to ensure high accuracy we employ reconstruction spaces consisting of splines or (piecewise) polynomials. We analyze the relation between the dimension of the reconstruction space and the bandwidth of the nonuniform samples, and show that it is linear for splines and piecewise polynomials of fixed degree, and quadratic for piecewise polynomials of varying degree.