Choosing the points in product integration

A product-integration rule for the integral ∫/sup b//sub a/k (t) f (t) dt is a rule of the form summation/sup n//sub i/=1w/sub i/ f (t/sub i/), with the weights w1,...,w/sub n/ chosen so that the rule is exact if f is any linear combination of a chosen set of functions phi1,...,phi/sub n/. For some choices of ]phi/sub j/], including the polynomial case, the points ]t/sub i/] need to be carefully chosen if reliable results are to be obtained. In this paper known convergence results for the polynomial case with well-chosen points are summarized and illustrated, and extended to some nonpolynomial cases, including one proposed by Y.E. Kim for use in solving the three-body Faddeev equations. The convergence theorems yield practical prescriptions for choosing the points ]t/sub i/]