快速旋转相位函数的数值积分
Numerical integration of functions of a rapidly rotating phase
摘要：
我们提出了一种算法的有效的数值计算积分的形式\[ I(ω) = \int_0^1 F( x,\mathrm e^{\mathrm i ωx}; ω) \, \mathrm d x \]足够光滑，但在其他方面任意$F$和$ω\gg 1$。该方法完全是“黑箱”，即，不需要力矩积分的显式计算或涉及$F$的其他预先计算。其性能在频率$ω$上是一致的。当$F$是解析的时，我们证明了该方法关于其阶指数收敛，并给出了其误差特性的数值演示。
英文摘要：
We present an algorithm for the efficient numerical evaluation of integrals of the form \[ I(ω) = \int_0^1 F( x,\mathrm e^{\mathrm i ωx}; ω) \, \mathrm d x \] for sufficiently smooth but otherwise arbitrary $F$ and $ω\gg 1$. The method is entirely "black-box", i.e., does not require the explicit computation of moment integrals or other pre-computations involving $F$. Its performance is uniform in the frequency $ω$. We prove that the method converges exponentially with respect to its order when $F$ is analytic and give a numerical demonstration of its error characteristics.
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