N derivatives are necessary for order N+1 convergence in quadrature: a converse result

Abstract

Results on the error bounds of quadrature methods are well known - most state that if the method has degree N, and the integrand has N derivatives, then the error is order N+1. We prove here a converse: that if the integrand fails to have N derivatives, even only at a finite number of points, no method, regardless of its degree, can guarantee convergence more than order N. Even if the integrand fails to have N derivatives at just 3 (for even N, 2) points, no method can produce order more than N+1 convergence. This is done by an adversarial proof: we explicitly construct the functions that exhibit such error; simple splines turn out to suffice.