A Bayesian framework for spectral reprojection

TL;DR

This paper introduces a Bayesian approach to spectral reprojection that enhances robustness against noise, improves parameter selection, and quantifies uncertainty in the reconstruction of non-periodic functions.

Contribution

It presents a novel Bayesian framework for spectral reprojection, addressing noise sensitivity, parameter tuning, and uncertainty quantification in Gegenbauer polynomial-based reconstructions.

Findings

Improved robustness to noise in spectral reprojection.

Enhanced parameter selection for Gegenbauer polynomials.

Quantification of uncertainty in the reconstructed solution.

Abstract

Fourier partial sum approximations yield exponential accuracy for smooth and periodic functions, but produce the infamous Gibbs phenomenon for non-periodic ones. Spectral reprojection resolves the Gibbs phenomenon by projecting the Fourier partial sum onto a Gibbs complementary basis, often prescribed as the Gegenbauer polynomials. Noise in the Fourier data and the Runge phenomenon both degrade the quality of the Gegenbauer reconstruction solution, however. Motivated by its theoretical convergence properties, this paper proposes a new Bayesian framework for spectral reprojection, which allows a greater understanding of the impact of noise on the reprojection method from a statistical point of view. We are also able to improve the robustness with respect to the Gegenbauer polynomials parameters. Finally, the framework provides a mechanism to quantify the uncertainty of the solution estimate.