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

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 $\lambda > - 1/2$, \[ a_{v,n}^{(\lambda )G} = \frac {\pi }{{n + \lambda }}\;{\sin ^{2\lambda }}\theta _{v,n}^{(\lambda )}\left \{ {1 - \frac {{\lambda (1 - \lambda )}}{{2{{(n + \lambda )}^2}{{\sin }^2}\theta _{v,n}^{(\lambda )}}} - \frac {{\lambda (1 - \lambda )\;[3(\lambda + 1)(\lambda - 2) + 4{{\sin }^2}\theta _{v,n}^{(\lambda )}]}}{{8{{(n + \lambda )}^4}{{\sin }^4}\theta _{v,n}^{(\lambda )}}}} \right \} + O({n^{ - 7}}).\]