Exclusive robustness of Gegenbauer method to truncated convolution errors

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