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A dynamical approach to accelerating numerical integration with equidistributed points. (English) Zbl 1166.65011
Proc. Steklov Inst. Math. 256, 275-289 (2007) and Tr. Mat. Inst. Steklova 256 (2007).
The paper is interesting and well written.
The authors show how the rate of convergence of the approximate integral with respect to Lebesgue measure can be significantly accelerated for the case of a real analytic function by considering linear combinations of some equidistributions (Theorem 1); the rate of convergence is better than that of any Newton-Cotes rule.
An analogue of Theorem 1 for the $$d$$-dimensional Lebesgue measure is also presented (Theorem 2).

##### MSC:
 65D32 Numerical quadrature and cubature formulas 37N30 Dynamical systems in numerical analysis 26E05 Real-analytic functions 41A55 Approximate quadratures
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