Abstract
This is a tutorial paper that gives the complete proof of a result of Frolov (Dokl Akad Nauk SSSR 231:818–821, 1976, [4]) that shows the optimal order of convergence for numerical integration of functions with bounded mixed derivatives. The presentation follows Temlyakov (J Complex 19:352–391, 2003, [13]), see also Temlyakov (Approximation of periodic functions, 1993, [12]).
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Notes
- 1.
A polynomial P is called irreducible over \(\mathbb {Q}\) if \(P=GH\) for two polynomials G, H with rational coefficients implies that one of them has degree zero. This implies that all roots of P must be irrational. In fact, every polynomial of the form \(\prod _{j=1}^d(x-b_j)-1\) with different \(b_j\in {\mathbb {Z}}\) is irreducible, but has not necessarily d different real roots.
- 2.
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Ullrich, M. (2016). On “Upper Error Bounds for Quadrature Formulas on Function Classes” by K.K. Frolov. In: Cools, R., Nuyens, D. (eds) Monte Carlo and Quasi-Monte Carlo Methods. Springer Proceedings in Mathematics & Statistics, vol 163. Springer, Cham. https://doi.org/10.1007/978-3-319-33507-0_31
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