  We discuss the reconstruction of piecewise smooth data from its (pseudo-)
spectral information. Spectral projections enjoy superior resolution provided
the data is globally smooth, while the presence of jump discontinuities is
responsible for spurious ${\cal O}(1)$ Gibbs oscillations in the neighborhood
of edges and an overall deterioration to the unacceptable first-order
convergence rate. The purpose is to regain the superior accuracy in the
piecewise smooth case, and this is achieved by mollification.
  Here we utilize a modified version of the two-parameter family of spectral
mollifiers introduced by Gottlieb & Tadmor [GoTa85]. The ubiquitous
one-parameter, finite-order mollifiers are based on dilation. In contrast, our
mollifiers achieve their high resolution by an intricate process of high-order
cancelation. To this end, we first implement a localization step using edge
detection procedure, [GeTa00a, GeTa00b]. The accurate recovery of piecewise
smooth data is then carried out in the direction of smoothness away from the
edges, and adaptivity is responsible for the high resolution. The resulting
adaptive mollifier greatly accelerates the convergence rate, recovering
piecewise analytic data within exponential accuracy while removing spurious
oscillations that remained in [GoTa85]. Thus, these adaptive mollifiers offer a
robust, general-purpose ``black box'' procedure for accurate post processing of
piecewise smooth data.

Numerical Analysis