Sinc Function and Gauss-Legendre quadrature

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:

$\frac{d^n}{d\omega^n} H(\omega) \Biggr|_{\omega=0}= 2\,\frac{d^n}{d\omega^n} \text{sinc}(\omega) \Biggr|_{\omega=0}=\frac{2\,(-1)^{n}}{n+1}$

where $n=0,2,4,\dots,2k$ is polynomial degree and $H(\omega)$ is frequency response of a quadrature rule.

This relation is very interesting as it allows us to build numerical integrators in frequency domain (see differentiators and smoothers for example).

How these two could possibly relate to each other? I had no idea, until I saw these two plots today:
sinc_gl
Green curve is a frequency response of Gauss-Legendre quadrature of order $N=15$ , red – $N=25$ and blue curve is the $\mathrm{sinc}(\omega)$ function.

As strange as it might seem, frequency response of GL quadrature coincides exactly with $\mathrm{sinc}(\omega)$ near zero and tend to be the one as order of quadrature grow. In other words, GL approximates the $\mathrm{sinc}(\omega)$ in frequency domain.

How cool is this? Just look how different the initial pre-images are:
sinc_nodes

Fourier transform of GL nodes (dots) approximate the Fourier transform of a blue square (which is $=\mathrm{sinc}(\omega)$ ) near zero.

I love such deep inter-connections between seemingly disperse fields of mathematics. First it should be like this (consequence of universal laws of nature), secondly it makes me believe that we are using the right instruments to describe the nature (which make inter-connections apparent).

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Sinc Function and Gauss-Legendre quadrature

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