Asymptotic representation of the weights in the Gauss quadrature formula
Consider the quadrature formula of Gauss-Jacobi for Legendre's weight dx on [0,1]. Given n, let \(\lambda_{n,j}\), \(j=1,2,...,n\), be the Christoffel coefficient corresponding to the zero \(x_{n,j}\), \(j=1,2,...,n\), of the nth Legendre orthogonal polynomial. Set cos \(\theta\) \({}_{n,j}=x_{n,j}\). It is proved that \[ \lambda_{n,j}=\frac{\pi}{2n+1}\sin \theta_{n,j}+O(\frac{1}{n^ 3\sin \theta_{n,j}}). \]
scientific article; zbMATH DE number 4212703