On the convergence of interpolatory product integration rules based on Gauss, Radau and Lobatto points (Q1089549)

Interpolatory product integration is concerned with numerical integration rules of the form \[ Q_ n(f;k)=\sum^{m_ n}_{i=\ell_ n}w_{ni}(k)f(x_{ni}) \] to the integral \(I(kf)=\int^{b}_{a}k(x)f(x)dx\), where \(k\in L_ 1(a,b)\). The nodes \(X=\{x_{ni}\}\) are specified in advance, and the coefficients are chosen so that there is no truncation error when f is a polynomial of degree \(\leq m_ n-\ell_ n\). Such formulas are important for the numerical solution of the Fredholm integral equation. The author gives sufficient conditions which insure the convergence \(\lim_{n\to \infty}Q_ n(f;k)=I(kf)\) for all Riemann integrable functions f for the nodes corresponding to the Gauss, Radau and Lobatto formulas, with respect to a sufficient regular k. He calls such k a generalized smooth Jacobi weight function. Much of his proofs are modelled on a theorem by \textit{I. H. Sloan} and \textit{W. E. Smith} [SIAM J. Numer. Anal. 19, 427-442 (1982; Zbl 0491.41002)].