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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

On multiple node Gaussian quadrature formulae


Author: David L. Barrow
Journal: Math. Comp. 32 (1978), 431-439
MSC: Primary 41A55; Secondary 65D32
MathSciNet review: 482257
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Abstract: Let $ {\mu _1}, \ldots ,{\mu _k}$ be odd positive integers and $ n = \Sigma _{i = 1}^k({\mu _i} + 1)$. Let $ \{ {\mu _i}\} _{i = 1}^n$ be an extended Tchebycheff system on $ [a,b]$. Let L be a positive linear functional on $ U \equiv {\operatorname{span}}(\{ {\mu _i}\} )$. We prove that L has a unique representation in the form

$\displaystyle L(p) = \sum\limits_{i = 1}^k {\sum\limits_{j = 0}^{{\mu _i} - 1} {{a_{ij}}{p^{(j)}}({t_i}),\quad a < {t_1} < \cdots < {t_k} < b,} } $

for all $ p \in U$. The proof uses the topological degree of a mapping $ F:\overline D \subset {R^k} \to {R^k}$. The result is proved by showing that the equation $ F(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} ) = 0$ has a unique solution, which in turn is proved by showing that F has degree 1 and that for any solution $ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} $ to the equation $ F(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} ) = 0$, $ \det F\prime(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} ) > 0$. We also give extensions to the cases when the $ \{ {u_i}\} $ are a periodic extended Tchebycheff system and when L is a nonnegative linear functional.

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DOI: http://dx.doi.org/10.1090/S0025-5718-1978-0482257-0
PII: S 0025-5718(1978)0482257-0
Article copyright: © Copyright 1978 American Mathematical Society