Asymptotic behavior of Sobolev-type orthogonal polynomials on the unit circle

We study the asymptotic behavior of the sequence of polynomials orthogonal with respect to the discrete Sobolev inner product on the unit circle. < f, g > = integral(0)(2 pi) f(e(i theta)) <(g(e(i theta)))over bar>d theta(0) + f(Z) Ag(Z)(H), where f(Z)=(f(z(1)), ...,f((l1))(z(1)), ..., f(z(m)), ...,f((lm))(z(m))), A is a M x M positive definite matrix or a positive semidefinite diagonal block matrix, M = l(1) + ... + l(m) + m, d mu belongs to a certain class of measures, and \z(i)\ > 1, i = 1,2, ..., m. (C) 1999 Academic Press.