Optimal filter and mollifier for piecewise smooth spectral data

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.