= Exponential convergence of Gaussian quadrature
{title2=$|I-Q_n|\leq C e^{-Ln}$}
For positive <quadrature weights> summing to the total mass $M$, a <quadrature rule> exact on <polynomials> through <polynomial degree> $d$ satisfies $|I(f)-Q(f)|\leq2M\|f-p_d\|_\infty$ 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 $|c_j|\leq e^{-Kj}$.
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