On the variance of gaussian quadrature formulae in the ultraspherical case |
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Authors: | K J Förster K Petras |
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Affiliation: | 1. Institut für Mathematik, Universit?t Hildesheim, 31141, Hildesheim, Germany 2. Institut für Angewandte Mathematik, Technische Universit?t Braunschweig, 38106, Braunschweig, Germany
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Abstract: | For ultraspherical weight functions ωλ(x)=(1–x2)λ–1/2, we prove asymptotic bounds and inequalities for the variance Var(Q n G ) of the respective Gaussian quadrature formulae Q n G . A consequence for a large class of more general weight functions ω and the respective Gaussian formulae is the following asymptotic result, $$\mathop {lim}\limits_{n \to \infty } n \cdot Var\left( {Q_n^G } \right) = \pi \int_{ - 1}^1 {\omega ^2 \left( x \right)\sqrt {1 - x^2 } dx.} $$ |
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