On the existence of orthogonal polynomials for oscillatory weights on a bounded interval

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.