Abstract
Asymptotic expansions for oscillatory integrals typically depend on the values and derivatives of the integrand at a small number of critical points. We show that using values of the integrand at certain complex points close to the critical points can actually yield a higher asymptotic order approximation to the integral. This superinterpolation property has interesting ramifications for numerical methods based on exploiting asymptotic behaviour. The asymptotic convergence rates of Filon-type methods can be doubled at no additional cost. Numerical steepest descent methods already exhibit this high asymptotic order, but their analyticity requirements can be significantly relaxed. The method can be applied to general oscillators with stationary points as well, through a simple change of variables.
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Communicated by Arieh Iserles.
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Huybrechs, D., Olver, S. Superinterpolation in Highly Oscillatory Quadrature. Found Comput Math 12, 203–228 (2012). https://doi.org/10.1007/s10208-011-9102-8
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DOI: https://doi.org/10.1007/s10208-011-9102-8