Stable reconstructions in Hilbert spaces and the resolution of the Gibbs phenomenon

TL;DR

This paper presents a stable Hilbert space reconstruction method using finite samples and tailored bases, effectively addressing the Gibbs phenomenon by achieving high accuracy with Fourier coefficients.

Contribution

It introduces a novel stable reconstruction technique in Hilbert spaces that improves approximation accuracy, especially for piecewise analytic functions, by choosing suitable bases.

Findings

Stable reconstruction is achievable with finite samples and appropriate basis choice.

The method effectively mitigates the Gibbs phenomenon for Fourier-based reconstructions.

Numerical examples demonstrate superior accuracy compared to existing methods.

Abstract

We introduce a method to reconstruct an element of a Hilbert space in terms of an arbitrary finite collection of linearly independent reconstruction vectors, given a finite number of its samples with respect to any Riesz basis. As we establish, provided the dimension of the reconstruction space is chosen suitably in relation to the number of samples, this procedure can be numerically implemented in a stable manner. Moreover, the accuracy of the resulting approximation is completely determined by the choice of reconstruction basis, meaning that the reconstruction vectors can be tailored to the particular problem at hand. An important example of this approach is the accurate recovery of a piecewise analytic function from its first few Fourier coefficients. Whilst the standard Fourier projection suffers from the Gibbs phenomenon, by reconstructing in a piecewise polynomial basis, we obtain an approximation with root exponential accuracy in terms of the number of Fourier samples and exponential accuracy in terms of the degree of the reconstruction function. Numerical examples illustrate the advantage of this approach over other existing methods.