On choosing the points in product integration

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Abstract
A product‐integration rule for the integral Fbak (t) f (t) dt is a rule of the form Jni=1wi f (ti), with the weights w1,...,wn chosen so that the rule is exact if f is any linear combination of a chosen set of functions φ1,...,φn . For some choices of {φj}, including the polynomial case, the points {ti} 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 {ti} .

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