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Asymptotic representation of the weights in the Gauss quadrature formula. (English. Russian original) Zbl 0733.41033
Math. Notes 47, No. 4, 354-358 (1990); translation from Mat. Zametki 47, No. 4, 63-68 (1990).
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 \[ \lambda_{n,j}=\frac{\pi}{2n+1}\sin \theta_{n,j}+O(\frac{1}{n^ 3\sin \theta_{n,j}}). \]
41A55 Approximate quadratures
Full Text: DOI
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