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Mathematics of Computation

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On estimates for the weights in Gaussian quadrature in the ultraspherical case

Authors: Klaus-Jürgen Förster and Knut Petras
Journal: Math. Comp. 55 (1990), 243-264
MSC: Primary 65D32; Secondary 41A55
MathSciNet review: 1023758
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Abstract: In this paper the Christoffel numbers $ a_{v,n}^{(\lambda )G}$ 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 _{v,n}^{(\lambda )}$ the trigonometric representation of the Gaussian nodes, we obtain for $ \lambda \in [0,1]$ the inequalities

\begin{displaymath}\begin{array}{*{20}{c}} {\frac{\pi }{{n + \lambda }}{{\sin }^... ...{\sin }^{2\lambda }}\theta _{v,n}^{(\lambda )}} \\ \end{array} \end{displaymath}

and similar results for $ \lambda \notin (0,1)$. Furthermore, assuming that $ \theta _{v,n}^{(\lambda )}$ remains in a fixed closed interval, lying in the interior of $ (0,\pi )$ as $ n \to \infty $, we show that, for every fixed $ \lambda > - 1/2$,

$\displaystyle a_{v,n}^{(\lambda )G} = \frac{\pi }{{n + \lambda }}\;{\sin ^{2\la... ...\lambda )}^4}{{\sin }^4}\theta _{v,n}^{(\lambda )}}}} \right\} + O({n^{ - 7}}).$

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Keywords: Christoffel numbers, Gaussian quadrature
Article copyright: © Copyright 1990 American Mathematical Society