Adaptive filters for piecewise smooth spectral data

We introduce a new class of exponentially accurate filters for processing piecewise smooth spectral data. Our study is based on careful error decompositions, focusing on a rather precise balance between physical space localization and the usual moments condition. Exponential convergence is recovered by optimizing the order of the filter as an adaptive function of both the projection order and the distance to the nearest discontinuity. Combined with the automated edge detection methods, e.g. Gelb & Tadmor (2002, Math. Model. Numer. Anal., 36, 155-175)., adaptive filters provide a robust, computationally efficient, black box procedure for the exponentially accurate reconstruction of a piecewise smooth function from its spectral information.