Error Bounds for the Numerical Integration of Functions with Limited Smoothness

TL;DR

This paper analyzes error bounds for numerical integration methods like Gauss and Clenshaw-Curtis quadrature when applied to functions with limited smoothness, providing a broader theoretical framework for understanding their performance.

Contribution

It extends existing error bound results to a wide class of quadrature formulas and weight functions, clarifying their behavior for less smooth integrands.

Findings

Error bounds can be generalized to many quadrature formulas.

Limited smoothness affects convergence rates.

Theoretical results align with observed numerical behavior.

Abstract

Recently, Trefethen (SIAM Review 50 (2008), 67--87) and Xiang and Bornemann (SIAM J. Numer. Anal. 50 (2012), 2581--2587) investigated error bounds for n-point Gauss and Clenshaw-Curtis quadrature for the Legendre weight with integrands having limited smoothness properties. Putting their results into the context of classical quadrature theory, we find that the observed behaviour is by no means surprising and that it can essentially be proved for a very large class of quadrature formulas with respect to a broad set of weight functions.