Large degree asymptotics of orthogonal polynomials with respect to an oscillatory weight on a bounded interval

Abstract We consider polynomials p n ω ( x ) that are orthogonal with respect to the oscillatory weight w ( x ) = e i ω x on [ − 1 , 1 ] , where ω > 0 is a real parameter. A first analysis of p n ω ( x ) for large values of ω was carried out in Asheim et al. (2014), in connection with complex Gaussian quadrature rules with uniform good properties in ω . In this contribution we study the existence, asymptotic behavior and asymptotic distribution of the roots of p n ω ( x ) in the complex plane as n → ∞ . The parameter ω grows with n linearly. The tools used are logarithmic potential theory and the S -property, together with the Riemann–Hilbert formulation and the Deift–Zhou steepest descent method.