Zur Struktur von Quadraturformeln |
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Authors: | Dr Franz Locher |
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Institution: | (1) Mathematisches Institut der Universität Tübingen, Brunnenstraße 27, D-7400 Tübingen, Bundesrepublik Deutschland |
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Abstract: | Summary Usually the errorR
n
(j) of a quadrature formula is estimated with the aid of theL
1-norm of the Peano kernel. It is shown that this term may be estimated rather sharp using the norm Q
n
of the quadrature rule. Then it follows that formulas with non-negative weights are favourable also in the sense of minimizing theL
1-norm of the kernel. A remainder term of the typeR
n
(f)=cf(n+1) ( ) is possible iff the kernel is definite. In the case of an interpolatory formula this definiteness is usually shown by an application of the so-called V-method . We determine the optimal formulas in the sense of this method. Then we analyse the influence of the structure of the mesh on the norm of a formula. We find that on an equidistant mesh withm nodes there exists a rule with a small norm if the order is not greater than
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Keywords: | |
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