Multiple integration over bounded and unbounded regions

摘要

A method is proposed for integrating over Rn functions that are reasonably smooth and rapidly decaying at infinity. The method makes use of an infinite lattice of quadrature points, truncated at some suitable radius. In certain circumstances it is shown that the ‘best’ lattice, among all those with a given determinant, is the one whose dual lattice corresponds to the densest sphere packing. The method of Sag and Szekeres for integration over n-dimensional spheres and cubes is shown to be a special case, with a lattice that is not necessarily optimal.