The Levin method is a well-known technique for evaluating oscillatory integrals, which operates by solving a certain ordinary differential equation in order to construct an antiderivative of the integrand. It was long believed that this approach suffers from "low-frequency breakdown," meaning that the accuracy of the calculated value of the integral deteriorates when the integrand is only slowly oscillating. Recently presented experimental evidence, however, suggests that if a Chebyshev spectral method is used to discretize the differential equation and the resulting linear system is solved via a truncated singular value decomposition, then no low-frequency breakdown occurs. Here, we provide a proof that this is the case, and our proof applies not only when the integrand is slowly oscillating, but even in the case of stationary points. Our result puts adaptive schemes based on the Levin method on a firm theoretical foundation and accounts for their behavior in the presence of stationary points. We go on to point out that by combining an adaptive Levin scheme with phase function methods for ordinary differential equations, a large class of oscillatory integrals involving special functions, including products of such functions and the compositions of such functions with slowly-varying functions, can be easily evaluated without the need for symbolic computations. Finally, we present the results of numerical experiments which illustrate the consequences of our analysis and demonstrate the properties of the adaptive Levin method.

翻译：莱文方法是评估振荡积分的一种著名技术，其运作原理是通过求解特定常微分方程来构造被积函数的反导数。长期以来，人们认为该方法存在“低频崩溃”问题，即当被积函数仅缓慢振荡时，积分计算值的精度会下降。然而，近期实验证据表明，若采用切比雪夫谱方法离散化该微分方程，并通过截断奇异值分解求解所得线性系统，则不会出现低频崩溃现象。本文对此提供严格证明，该证明不仅适用于被积函数缓慢振荡的情形，还涵盖驻点情况。此结论为基于莱文方法的自适应方案奠定了坚实的理论基础，并解释了其在驻点存在时的行为表现。进一步地，我们指出：通过将自适应莱文方案与常微分方程的相位函数方法相结合，可以轻松评估涉及特殊函数（包括此类函数的乘积及其与缓变函数的复合函数）的大类振荡积分，且无需符号计算。最后，我们展示数值实验结果，这些结果既印证了理论分析的有效性，也揭示了自适应莱文方法的特性。