  Spectral methods provide an elegant and efficient way of numerically solving
differential equations of all kinds. For smooth problems, truncation error for
spectral methods vanishes exponentially in the infinity norm and $L_2$-norm.
However, for non-smooth problems, convergence is significantly worse---the
$L_2$-norm of the error for a discontinuous problem will converge at a
sub-linear rate and the infinity norm will not converge at all. We explore and
improve upon a post-processing technique---optimally convergent mollifiers---to
recover exponential convergence from a poorly-converging spectral
reconstruction of non-smooth data. This is an important first step towards
using these techniques for simulations of realistic systems.
