On the variance of gaussian quadrature formulae in the ultraspherical case

Abstract

For ultraspherical weight functions ωλ(x)=(1–x2)λ–1/2, we prove asymptotic bounds and inequalities for the variance Var(QnG) of the respective Gaussian quadrature formulae QnG. A consequence for a large class of more general weight functions ω and the respective Gaussian formulae is the following asymptotic result, $$\mathop {lim}\limits_{n \to \infty } n \cdot Var\left( {Q_n^G } \right) = \pi \int_{ - 1}^1 {\omega ^2 \left( x \right)\sqrt {1 - x^2 } dx.} $$