We present a method for the efficient approximation of integrals with highly oscillatory vector-valued kernels, such as integrals involving Airy functions or Bessel functions. We construct a vector-valued version of the asymptotic expansion, which allows us to determine the asymptotic order of a Levin-type method. Levin-type methods are constructed using collocation, and choosing a basis based on the asymptotic expansion results in an approximation with significantly higher asymptotic order.
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Olver, S. Numerical approximation of vector-valued highly oscillatory integrals . Bit Numer Math 47, 637–655 (2007). https://doi.org/10.1007/s10543-007-0137-9
- Highly oscillatory integrals, asymptotics, quadrature