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.

翻译：傅里叶部分和逼近为光滑周期函数提供了指数级精度，但对非周期函数却会产生著名的吉布斯现象。谱重投影通过将傅里叶部分和投影到吉布斯互补基（通常指定为盖根鲍尔多项式）上来解决吉布斯现象。然而，傅里叶数据中的噪声和龙格现象都会降低盖根鲍尔重建解的质量。受其理论收敛性质的启发，本文提出了一种新的谱重投影贝叶斯框架，该框架允许从统计角度更深入地理解噪声对重投影方法的影响。我们还能够提高方法对盖根鲍尔多项式参数的鲁棒性。最后，该框架提供了一种量化解估计不确定性的机制。