Ratio and relative asymptotics of polynomials orthogonal on an arc of the unit circle

Abstract

Ratio and relative asymptotics are given for sequences of polynomials orthogonal with respect to measures supported on an arc of the unit circle, where their absolutely continuous component is positive almost everywhere. The results obtained extend to this setting known ones given by Rakhmanov and Máté, Nevai, and Totik for the case when the arc is the whole unit circle. Technically speaking, the main feature is the use of orthogonality with respect to varying measures.