Summary: We consider polynomials orthogonal with respect to the linear functional \(\mathcal L : \mathcal P\to \mathbb C\), defined by \(\mathcal L[p] = \int _{-1}^1 p(x)(1-x^2)^{\lambda -1/2}\exp(\text i \zeta x)\text dx\), where \(\mathcal P\) is a linear space of all algebraic polynomials, \(\lambda > - 1/2\) and \(\zeta \in \mathbb R\). We prove the existence of such polynomials for some pairs of \(\lambda \) and \(\zeta \), give some their properties, and finally give an application to numerical integration of highly oscillatory functions.