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This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.On Estimates for the Weights in Gaussian Quadrature in the Ultraspherical Case
KlausJürgen Förster and Knut Petras
Mathematics of Computation
Vol. 55, No. 191 (Jul., 1990), pp. 243264
Published by: American Mathematical Society
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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Mathematics of Computation © 1990 American Mathematical Society