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

It is shown that the polynomials orthogonal on (−1,1) w.r.t. the oscillatory weight eiωx exist if ω is a transcendental number and tan⁡ω/ω∈Q. Also, it is proved that such orthogonal polynomials exist for almost all ω>0, and the roots are all simple if ω>0 is either small enough or large enough.