Abstract
We consider Gaussian quadrature formulae Q, n GIN, approximating the integral\(I({\text{f}}): = \smallint _{ - 1}^1w({\text{x}})f({\text{x}})dx.\) Let \(f({\text{z}}) = \mathop \Sigma \limits_{i = 0}^\infty \alpha _i^f{z^i}\) be analytic in \({K_r}: = \{ {\text{z}} \in \mathbb{C}:|{\text{z}}| < {\text{r}}\} ,{\text{r}} > 1\) and \(|{\text{f}}{|_{\text{r}}}{\text{: = sup\{ |}}\alpha _{\text{i}}^{\text{f}}{\text{|}}{{\text{r}}^{\text{i}}}{\text{:i}} \in {\mathbb{N}_ \circ }{\text{,}}{{\text{R}}_{\text{n}}}{\text{(}}{{\text{q}}_{\text{i}}}{\text{)}} \ne {\text{0\} < }}\infty \) where Rn = I -Qn is the error functional and qi (x):= x1. Rn is continuous with respect to | • |r and \(||{\text{Rn|| = }}\mathop \Sigma \limits_{{\text{i = 0}}}^\infty {\text{[|}}{{\text{R}}_{\text{n}}}{\text{(}}{{\text{q}}_{\text{i}}}{\text{)|/}}{{\text{r}}^{\text{i}}}{\text{]}}\) holds. For \({\text{w(x) = }}\frac{1}{{{\text{c - }}{{\text{x}}^{\text{2}}}}}{\text{(1 - x)}}\alpha {\text{(1 + x)}}\beta {\text{ = }} \pm \frac{1}{2},c > 1,\) we explicitly calculate the norm of Rn
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© 1985 Birkhäuser Verlag Basel
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Akrivis, G. (1985). Die Fehlernorm Spezieller Gauss-Quadraturformeln. In: Hämmerlin, G., Hoffmann, KH. (eds) Constructive Methods for the Practical Treatment of Integral Equations. International Series of Numerical Mathematics, vol 73. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9317-6_1
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DOI: https://doi.org/10.1007/978-3-0348-9317-6_1
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