Piecewise Smooth Data from its Spectral Information

We discuss the reconstruction of piecewise smooth data from its (pseudo-) spectral information. Spectral projections enjoy superior resolution provided the data is globally smooth, while the presence of jump discontinuities is responsible for spurious O(1) Gibbs oscillations in the neighborhood of edges and an overall deterioration to the unacceptable rst-order convergence rate. The purpose is to regain the superior accuracy in the piecewise smooth case, and this is achieved by molli cation. Here we utilize a modi ed version of the two-parameter family of spectral molli ers introduced by Gottlieb & Tadmor [GoTa85]. The ubiquitous one-parameter, nite-order molli ers are based on dilation. In contrast, our molli ers achieve their high resolution by an intricate process of high-order cancelation. To this end, we rst implement a localization step using edge detection procedure, [GeTa00a, GeTa00b]. The accurate recovery of piecewise smooth data is then carried out in the direction of smoothness away from the edges, and adaptivity is responsible for the high resolution. The resulting adaptive molli er greatly accelerates the convergence rate, recovering piecewise analytic data within exponential accuracy while removing spurious oscillations that remained in [GoTa85]. Thus, these adaptive molli ers o er a robust, general-purpose \black box" procedure for accurate post processing of piecewise smooth data.