Spectral projections enjoy high order convergence for globally smooth functions. However, a single discontinuity introduces O(1) spurious oscillations near the discontinuity and reduces the high order convergence rate to first order, Gibbs' Phenomena. Although a direct expansion of the function in terms of its global moments yields this low order approximation, high resolution information is retained in the global moments. Two techniques for the resolution of the Gibbs' phenomenon are discussed, filtering and reprojection methods. An adaptive filter with optimal joint time-frequency localization is presented, which recovers a function from its N term Fourier projection within the error bound \exp(-Nd(x)), where d(x) is the distance from the point being recovered to the nearest discontinuity. Symmetric filtering, however, must sacrifice accuracy when approaching a discontinuity. To overcome this limitation, Gegenbauer postprocessing was introduced by Gottlieb, Shu, et al, which recovers a function from its N term Fourier projection within the error bound \exp(-N). An extension of Gegenbauer postprocessing with improved convergence and robustness properties is presented, the robust Gibbs complements. Filtering and reprojection methods will be put in a unifying framework, and their properties such as robustness and computational cost contrasted. This research was conducted jointly with Eitan Tadmor and Anne Gelb.
