On the existence of nonoscillatory phase functions for second order ordinary differential equations in the high-frequency regime

We observe that solutions of a large class of highly oscillatory second order linear ordinary differential equations can be approximated using nonoscillatory phase functions. In addition, we describe numerical experiments which illustrate several implications of this fact. For example, that many special functions of great interest - such as the Bessel functions Jν and Yν - can be evaluated accurately using a number of operations which is O(1) in the order ν. The present paper is devoted to the development of an analytical apparatus. Numerical aspects of this work will be reported at a later date.