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

 

How slowly can quadrature formulas converge?


Authors: Peter R. Lipow and Frank Stenger
Journal: Math. Comp. 26 (1972), 917-922
MSC: Primary 65D30
MathSciNet review: 0319356
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Abstract: Let $ \{ {Q_n}\} _{n = 1}^\infty $ denote a sequence of quadrature formulas, $ {Q_n}(f) \equiv \sum _{i = 1}^{{k_n}}w_i^{(n)}f(x_i^{(n)})$, such that $ {Q_n}(f) \to \int_0^1 f (x)dx$ for all $ f \in C[0,1]$. Let $ 0 < \varepsilon < \frac{1}{4}$ and a sequence $ \{ {a_n}\} _{n = 1}^\infty $ be given, where $ {a_1} \geqq {a_2} \geqq {a_3} \geqq ...$, and where $ {a_n} \to 0$ as $ n \to \infty $. Then there exists a function $ f \in C[0,1]$ and a sequence $ \{ {n_k}\} _{k = 1}^\infty $ such that $ \vert f(x)\vert \leqq 2{a_1}/\vert(1 - 4)\vert$, and such that $ \int_0^1 f (x)dx - {Q_{{n_k}}}(f) = {a_{k,}}k = 1,2,3,...$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1972-0319356-4
PII: S 0025-5718(1972)0319356-4
Keywords: Quadrature rules, convergence
Article copyright: © Copyright 1972 American Mathematical Society