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Mathematics
Department Colloquium |
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We discuss the reconstruction of piecewise smooth data from (pseudo-) spectral information. Spectral projections enjoy superior resolution provided the data is globally smooth. The presence of jump discontinuities, however, is responsible for spurious O(1) 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. This is followed by a second step of mollification--- we present a two-parameter family of spectral mollifiers which recover the data between those edges with exponential accuracy. The ubiquitous one-parameter, finite-order
mollifiers are based on dilation. In contrast, our family of two-parameter
spectral mollifiers achieve their high resolution by an intricate process
of high-order cancelation. |