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On Estimates for the Weights in Gaussian Quadrature in the Ultraspherical Case

Klaus-Jürgen Förster and Knut Petras
Mathematics of Computation
Vol. 55, No. 191 (Jul., 1990), pp. 243-264
DOI: 10.2307/2008803
Stable URL: http://www.jstor.org/stable/2008803
Page Count: 22
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Abstract

In this paper the Christoffel numbers $a^{(\lambda)G}_{\nu, n}$ for ultraspherical weight functions $w_\lambda, w_\lambda(x) = (1 - x^2)^{\lambda - 1/2}$, are investigated. Using only elementary functions, we state new inequalities, monotonicity properties and asymptotic approximations, which improve several known results. In particular, denoting by $\theta^{(\lambda)}_{\nu, n}$ the trigonometric representation of the Gaussian nodes, we obtain for $\lambda \in \lbrack 0, 1\rbrack$ the inequalities $$\frac{\pi}{n + \lambda} \sin^{2\lambda} \theta^{(\lambda)}_{\nu, n} \Bigg\{1 - \frac{\lambda(1 - \lambda)}{2(n + \lambda)^2 \sin^2 \theta^{(\lambda)}_{\nu, n}}\Bigg\}$$ $$\leq a^{(\lambda)G}_{\nu, n} \leq \frac{\pi}{n + \lambda} \sin^{2\lambda} \theta^{(\lambda)}_{\nu, n}$$ and similar results for $\lambda \notin (0, 1)$. Furthermore, assuming that $\theta^{(\lambda)}_{\nu, n}$ remains in a fixed closed interval, lying in the interior of $(0, \pi)$ as $n \rightarrow \infty$, we show that, for every fixed $\lambda > -1/2$, \begin{equation*}\begin{split}a^{(\lambda)G}_{\nu, n} = \frac{\pi}{n + \lambda} \sin^{2\lambda} \theta^{(\lambda)}_{\nu, n} \bigg\{1 - \frac{\lambda(1 - \lambda)}{2(n + \lambda)^2 \sin^2 \theta^{(\lambda)}_ {\nu, n}} \\- \frac{\lambda(1 - \lambda)\lbrack 3(\lambda + 1)(\lambda - 2) + 4 \sin^2 \theta^{(\lambda)}_{\nu, n}\rbrack}{8(n + \lambda)^4 \sin^4 \theta^{(\lambda)}_{\nu, n}}\bigg\} + O(n^{-7}).\end{split}\end{equation*}

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