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

Abstract:

We consider an $n$-dimensional integration rule $R_t^{(n)}$ of degree $2t - 1$ and of hypercubic symmetry. We derive theorems which set a lower bound in terms of $n$ and $t$ 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.