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

Abstract

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