Finite-order integration weights can be dangerous

Finite-order weights have been introduced in recent years to describe the often occurring situation that multivariate integrands can be approximated by a sum of functions each depending only on a small subset of the variables. The aim of this paper is to demonstrate the danger of relying on this structure when designing lattice integration rules, if the true integrand has components lying outside the assumed finite-order function space. It does this by proving, for weights of order two, the existence of 3-dimensional lattice integration rules for which the worst case error is of order O(N−1/2), where N is the number of points, yet for which there exists a smooth 3-dimensional integrand for which the integration rule does not converge.