The convergence of interpolatory product integration rules (Q1093824)

The interpolatory product integration consists in approximating the integral \(I(kf)=\int^{1}_{-1}k(x)f(x)dx\) by a sequence of numerical integration rules of the form \(Q_ n^{(r,s)}f=\sum^{r+s}_{i=1- r}w_{in}(k)f(x_{in})\) where r,s\(\in \{0,1\}\) with \(- 1=x_{0n}<x_{1n}<...<x_{nn}<x_{n+1,n}=1\) and \(w_{in}(k)\) chosen such that \(Q_ n^{(r,s)}f=I(kf)\) if f is a polynomial of degree \(<n+r+s\). The author considers the convergence of \(Q_ n^{(r,s)}f\) to I(kf) for all \(f\in C[-1,1]\) in the case where the points \(x_{in}\) are the zeros of the orthogonal polynomials associated with a generalized smooth Jacobi weight w(x) and obtains the conditions on k and w(x) which insure the convergence.