Abstract 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 finiteorder
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¯½), 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.
Compact Integration Rules as a quadrature method with some applications
