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Media type:
E-Article
Title:
Error Estimates in Gaussian Quadrature for Functions of Bounded Variation
Contributor:
Forster, Klaus-Jurgen;
Petras, Knut
imprint:
Society for Industrial and Applied Mathematics, 1991
Published in:SIAM Journal on Numerical Analysis
Language:
English
ISSN:
0036-1429
Origination:
Footnote:
Description:
<p>In this paper, for quadrature formulae Q-n and their respective remainder terms R<sub>n</sub>, error estimates are considered for the form R<sub>n</sub>[ f ] ≤ Q<sub>V</sub>(R<sub>n</sub>) V(f), where V(f) is the total variation over the interval of integration of the function f, and Q<sub>V</sub>(R<sub>n</sub>) is the best possible error constant for R<sub>n</sub>, i.e.,<tex-math>$\mathfrak{Q}_V (R_n) = \sup \{|R_n\lbrack f \rbrack\| \mid V(f) \leq 1\}$</tex-math>. For this class BV of functions of low-order continuity, the quality of positive quadrature formulae Q<sub>n</sub>having high algebraic degree of exactness is investigated. In particular, fot he Gaussian formulae Q<sup>G</sup>
<sub>n</sub>, simple explicit values of their error constants Q (R<sup>G</sup>
<sub>n</sub>) are stated, and it is shown that these differ from those of the respective optimal formulae only by a factor of less than π / 2. This result is based on an improvement of the classical separation theorem of Chebyshev- Markov-Stieltjes.</p>