How slowly can quadrature formulas converge?
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- by Peter R. Lipow and Frank Stenger PDF
- Math. Comp. 26 (1972), 917-922 Request permission
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 $|f(x)| \leqq 2{a_1}/|(1 - 4)|$, and such that $\int _0^1 f (x)dx - {Q_{{n_k}}}(f) = {a_{k,}}k = 1,2,3,...$.References
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Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Math. Comp. 26 (1972), 917-922
- MSC: Primary 65D30
- DOI: https://doi.org/10.1090/S0025-5718-1972-0319356-4
- MathSciNet review: 0319356