  It is shown that the orthogonal polynomials, corresponding to the oscillatory
weight $e^{\im\omega x}$, exists if $\omega$ is a transcendental number and
$\tan\omega/\omega\in\Q$. Also, it is proved that such orthogonal polynomials
exist for almost every $\omega>0$, and the roots are all simple if $\omega>0$
is either small enough or large enough.
