Limits on the number of function evaluations required by certain high-dimensional integration rules of hypercubic symmetry

Abstract

We consider an n n -dimensional integration rule R t ( n ) R_t^{(n)} of degree 2 t − 1 2t - 1 and of hypercubic symmetry. We derive theorems which set a lower bound in terms of n n and t t on the number of function evaluations such a rule requires. These results apply to spaces of integration which have hypercubic symmetry. In certain cases this bound is very close to the number of points required by a known rule.