  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 important implications of this fact. For example, that many special
functions of great interest --- such as the Bessel functions $J_\nu$ and
$Y_\nu$ --- can be evaluated accurately using a number of operations which is
$O(1)$ in the order $\nu$. The present paper is devoted to the development of
an analytical apparatus. Numerical aspects of this work will be reported at a
later date.
