Spectral projections enjoy superior resolution provided the data is globally smooth.
The presence of jump discontinuities, however, is responsible for spurious O Gibbs oscillations in the neighborhood of such jumps, and an overall deterioration to the unacceptable first-order convergence rate of spectral projections. The purpose is to regain the superior exponential accuracy in the piecewise smooth case, and this is achieved in two complementing steps. First, a localization step using a novel detection procedure based on concentration kernels which identify both the location and amplitudes of finitely many edges. Accurate recovery of piecewise smooth data is carried out in the direction of smoothness, and adaptivity is responsible for the high resolution.