On the Convergence Rates of Gauss and Clenshaw-Curtis Quadrature for Functions of Limited Regularity

We study the optimal general rate of convergence of the n-point quadrature rules of Gauss and Clenshaw–Curtis when applied to functions of limited regularity: if the Chebyshev coefficients decay at a rate O(n−s−1) for some s > 0, Clenshaw–Curtis and Gauss quadrature inherit exactly this rate. The proof (for Gauss, if 0 < s < 2, there is numerical evidence only) is based on work of Curtis, Johnson, Riess, and Rabinowitz from the early 1970s and on a refined estimate for Gauss quadrature applied to Chebyshev polynomials due to Petras (1995). The convergence rate of both quadrature rules is up to one power of n better than polynomial best approximation; hence, the classical proof strategy that bounds the error of a quadrature rule with positive weights by polynomial best approximation is doomed to fail in establishing the optimal rate.