Exponential convergence of Gaussian quadrature ($|I-Q_n|\le C{e}^{-Ln}$)

For positive quadrature weights summing to the total mass $M$, a quadrature rule exact on polynomials through polynomial degree $d$ satisfies $|I(f)-Q(f)|\le 2M\Vert f-p_d{\Vert }_{\infty }$ for every such polynomial $p_d$. This follows by subtracting the exactly integrated $p_d$ and applying the triangle inequality to the integral and weighted sum. For Gauss-Legendre quadrature with $n+1$ nodes, $M=2$ and $d=2n+1$. Exponentially decaying Chebyshev series coefficients therefore yield a bound proportional to $e^{-2Kn}$ when $|c_j|\le e^{-Kj}$.