Asymptotic Behavior of Sobolev-Type Orthogonal Polynomials on the Unit Circle

Abstract

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〉=∫f(eiθ)g(eiθ)dμ(θ)+f(Z)Ag(Z)H, where f(Z)=(f(z1), …, f(l1)(z1), …, f(zm), …, f(lm)(zm)), A is a M×M positive definite matrix or a positive semidefinite diagonal block matrix, M=l1+…+lm+m, dμ belongs to a certain class of measures, and |zi|>1, i=1, 2, …, m.