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

TL;DR

This paper investigates the convergence rates of Gauss and Clenshaw-Curtis quadrature for functions with limited regularity, showing they match the decay of Chebyshev coefficients and outperform polynomial approximation.

Contribution

It establishes the optimal convergence rates of these quadrature rules for functions with specific Chebyshev coefficient decay, extending previous work and providing refined estimates.

Findings

Clenshaw-Curtis and Gauss quadrature inherit the Chebyshev coefficient decay rate.

Convergence rate is up to one power of n better than polynomial best approximation.

The classical proof strategy fails to establish the optimal rate for positive weight quadrature.

Abstract

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.