Robust reprojection methods for the resolution of the Gibbs phenomenon

The classical Gibbs phenomenon exhibited by global Fourier projections and interpolants can be resolved in smooth regions by reprojecting in a truncated Gegenbauer series, achieving high resolution recovery of the function up to the point of discontinuity. Unfortunately, due to the poor conditioning of the Gegenbauer polynomials, the method suffers both from numerical round-off error and the Runge phenomenon. In some cases the method fails to converge. Following the work in [D. Gottlieb, C.W. Shu, Atti Conv. Lincei 147 (1998) 39–48], a more general framework for reprojection methods is introduced here. From this insight we propose an additional requirement on the reprojection basis which ameliorates the limitations of the Gegenbauer reconstruction. The new robust Gibbs complementary basis yields a reliable exponentially accurate resolution of the Gibbs phenomenon up to the discontinuities.