Error Bounds for the Numerical Integration of Functions with Limited Smoothness

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.