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Optimal filter and mollifier for piecewise smooth spectral data

Author: Jared Tanner
Journal: Math. Comp. 75 (2006), 767-790
MSC (2000): Primary 41A25, 42A10, 42A16, 42A20, 42C25, 65B10, 65T40
Published electronically: January 23, 2006
MathSciNet review: 2196991
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Abstract: We discuss the reconstruction of piecewise smooth data from its (pseudo-) spectral information. Spectral projections enjoy superior resolution provided the function is globally smooth, while the presence of jump discontinuities is responsible for spurious $ {\mathcal O}(1)$ Gibbs' oscillations in the neighborhood of edges and an overall deterioration of the convergence rate to the unacceptable first order. Classical filters and mollifiers are constructed to have compact support in the Fourier (frequency) and physical (time) spaces respectively, and are dilated by the projection order or the width of the smooth region to maintain this compact support in the appropriate region. Here we construct a noncompactly supported filter and mollifier with optimal joint time-frequency localization for a given number of vanishing moments, resulting in a new fundamental dilation relationship that adaptively links the time and frequency domains. Not giving preference to either space allows for a more balanced error decomposition, which when minimized yields an optimal filter and mollifier that retain the robustness of classical filters, yet obtain true exponential accuracy.

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Additional Information

Jared Tanner
Affiliation: Department of Statistics, Stanford University, Stanford, California 94305-9025

Keywords: Fourier series, filters, time-frequency localization, piecewise smooth, spectral projection
Received by editor(s): May 19, 2004
Received by editor(s) in revised form: January 29, 2005
Published electronically: January 23, 2006
Additional Notes: The author was supported in part by NSF Grants DMS 01-35345 and 04-03041.
Dedicated: This paper is dedicated to Eitan Tadmor for his direction
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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