A Dynamical Approach to Accelerating Numerical Integration with Equidistributed Points

We show how ideas originating in the theory of dynamical systems inspire a new approach to numerical integration of functions. Any Lebesgue integral can be approximated by a sequence of integrals with respect to equidistributions, i.e. evenly weighted discrete probability measures concentrated on an equidistributed set. We prove that, in the case where the integrand is real analytic, suitable linear combinations of these equidistributions lead to a significant acceleration in the rate of convergence of the approximate integral. In particular, the rate of convergence is faster than that of any Newton–Cotes rule.