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: [f(z), g (z)]s = integral(0)(2pi) f (e(itheta))<(g(e(iθ)))over bar> dmu(0)(theta) + integral(0)(2pi) f' (e(itheta)) <(g' (e(iθ)))over bar> dmu(1) (theta), z = e(itheta), with mu(0) a finite positive Borel measure on [0, 2,pi] and mu(1) a measure in the Szego's class. On the assumption that the Caratheodory function of mu(0) and the Szego function of mu(1) have analytic extension, we prove that the asymptotic formula holds true outside the disk and it can be extended inside the disk. (C) 2002 Elsevier Science Ltd. All rights reserved.