  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.
