  Spectral reconstructions provide rigorous means to remove the Gibbs
phenomenon and accelerate the convergence of spectral solutions in non-smooth
differential equations. In this paper, we show the concurrent emergence of
truncated convolution errors could entirely disrupt the performance of most
reconstruction techniques in the vicinity of discontinuities. They arise when
the Fourier coefficients of the product of two discontinuous functions, namely
$f=gh$, are approximated via truncated convolution of the corresponding Fourier
series, i.e. $\hat{f}_k\approx \sum_{|\ell|\leqslant
N}{\hat{g}_\ell\hat{h}_{k-\ell}}$. Nonetheless, we numerically illustrate and
rigorously prove that the classical Gegenbauer method remains exceptionally
robust against this phenomenon, with the reconstruction error still diminishing
proportional to $\mathcal{O}(N^{-1})$ for the Fourier order $N$, and
exponentially fast regardless of a constant. Finally, as a case study and a
problem of interest in grating analysis whence the phenomenon initially was
noticed, we demonstrate the emergence and practical resolution of truncated
convolution errors in grating modes, which constitute the basis of Fourier
modal methods.
