Strong asymptotics inside the unit disk for Sobolev orthogonal polynomials

In the present paper, we give sufficient conditions in order to establish the extension of the strong asymptotics up to the boundary and inside the unit disk for Sobolev orthogonal polynomials. We consider the following Sobolev inner product on the unit circle: with μ0 a finite positive Borel measure on [0,2π] and μ1 a measure in the Szeg ʹs class. On the assumption that the Carathéodory function of μ0 and the Szeg function ofμ1 have analytic extension, we prove that the asymptotic formula holds true outside the disk and it can be extended inside the disk.