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.Comment: 7 pages, the figure of the revision has an unsymmetric example, to
appear in SIAM J. Numer. Ana
