Peano kernels and bounds for the error constants
of Gaussian and related quadrature rules for Cauchy
principal value integrals |
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Authors: | Kai Diethelm |
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Institution: | Institut für Mathematik, Universit?t Hildesheim,
Marienburger Platz 22, D-31141 Hildesheim, Germany;
e-mail: diethelm@informatik.uni-hildesheim.de, DE
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Abstract: | Summary.
We show that, if
(),
the error term of
every modified positive interpolatory quadrature rule for
Cauchy principal value integrals of the type
,
, fulfills
uniformly for all
, and hence it is
of optimal
order of magnitude in the classes
().
Here, is a weight function with the property
.
We give explicit upper bounds for the Peano-type error
constants of such rules.
This improves and completes earlier results by
Criscuolo and Mastroianni
(Calcolo 22 (1985), 391–441 and Numer. Math.
54 (1989), 445–461)
and Ioakimidis (Math. Comp. 44 (1985), 191–198).
For the special case of the Gaussian rule, we
show that the restriction
can be dropped.
The results are based on a new representation of the
Peano kernels of these formulae via the Peano kernels of the underlying
classical quadrature formulae. This representation may also be
useful in connection with some different problems.
Received November 21, 1994 |
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Keywords: | Mathematics Subject Classification (1991): 65D30 41A55 65R10 |
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