Die Fehlernorm Spezieller Gauss-Quadraturformeln

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 R_n = I -Q_n is the error functional and q_i (x):= x¹. R_n 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 R_n