UPDATE: August 13, 2015. Finally I was able to see the relation between the two. Gauss-Legendre quadrature is based on exactness on polynomials, which can be re-formulated in terms of frequency domain. As it turned out, this condition has very simple form: where $n=0,2,4,\dots,2k$ is polynomial degree and $H(\omega)$ is frequency response of […]