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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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On estimates for the weights in Gaussian quadrature in the ultraspherical case
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by Klaus-Jürgen Förster and Knut Petras PDF
Math. Comp. 55 (1990), 243-264 Request permission

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 \[ \begin {array}{*{20}{c}} {\frac {\pi }{{n + \lambda }}{{\sin }^{2\lambda }}\theta _{v,n}^{(\lambda )}\left \{ {1 - \frac {{\lambda (1 - \lambda )}}{{2{{(n + \lambda )}^2}{{\sin }^2}\theta _{v,n}^{(\lambda )}}}} \right \}} \\ { \leq a_{v,n}^{(\lambda )G} \leq \frac {\pi }{{n + \lambda }}\;{{\sin }^{2\lambda }}\theta _{v,n}^{(\lambda )}} \\ \end {array} \] 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 $\lambda > - 1/2$, \[ a_{v,n}^{(\lambda )G} = \frac {\pi }{{n + \lambda }}\;{\sin ^{2\lambda }}\theta _{v,n}^{(\lambda )}\left \{ {1 - \frac {{\lambda (1 - \lambda )}}{{2{{(n + \lambda )}^2}{{\sin }^2}\theta _{v,n}^{(\lambda )}}} - \frac {{\lambda (1 - \lambda )\;[3(\lambda + 1)(\lambda - 2) + 4{{\sin }^2}\theta _{v,n}^{(\lambda )}]}}{{8{{(n + \lambda )}^4}{{\sin }^4}\theta _{v,n}^{(\lambda )}}}} \right \} + O({n^{ - 7}}).\]
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Additional Information
  • © Copyright 1990 American Mathematical Society
  • Journal: Math. Comp. 55 (1990), 243-264
  • MSC: Primary 65D32; Secondary 41A55
  • DOI: https://doi.org/10.1090/S0025-5718-1990-1023758-1
  • MathSciNet review: 1023758