A FORMULA FOR OPTIMAL INTEGRATION IN H-2

The weights a(j) of the optimal integration formula Q = E(j)a(j)f(z(j)) in H-2 for given integration points z(j) are the exact integrals of the cardinal functions in the corresponding formula for optimal evaluation. By writing these cardinal functions as sums of their principal values, we very easily obtain a closed formula for the weights. In the case of real z(j)'s, this formula makes explicit a series formula of Wilf. We compare numerically the accuracy of the optimal formula with that of some well-known integration formulae. For points equidistant on a circle of radius r, the formula allows an alternate derivation of a formula obtained by Golomb. We give also the barycentric formula for optimal evaluation with these points, as well as an experimentally stable sequence of radii r for integrating with an increasing number of points.