On the convergence rates of Gauss and Clenshaw-Curtis quadrature for functions of limited regularity

Details On the convergence rates of Gauss and Clenshaw-Curtis quadrature for functions of limited regularity On the convergence rates of Gauss and Clenshaw-Curtis quadrature for functions of limited regularity

2012-03-12

arxiv.org/abs/1203.2445v1

Mathematics

Numerical Analysis

7 pages, 1 figure

Scientific paper

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). There are cases where 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 would bound the error of a quadrature rule with positive weights by polynomial best approximation is doomed to fail in establishing the optimal rate.

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