  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.
