INTEGRATION OF NONPERIODIC FUNCTIONS OF 2 VARIABLES BY FIBONACCI LATTICE RULES

Two-dimensional lattice rules are applied to continuous functions over the unit square which do not have a continuous periodic extension, It is shown that, provided lattice points at vertices and edges are treated appropriately, certain functions (including all bilinear functions) are integrated exactly whenever the lattice contains a (possibly rotated) square unit cell. The Fibonacci lattice with denominators F(k) for the nodes is then shown to have a square unit cell if and only if k is odd. Numerical experiments for Fibonacci rules and copies of Fibonacci rules confirm that there are significant differences between the odd-k and even-k cases.