A family of Sobolev orthogonal polynomials on the unit circle

The aim of this paper is to study the polynomials orthogonal with respect to the following Sobolev inner product:〈P,Q〉s1=∫02πP(eiθ)Q(eiθ) dμ(θ)+1λ∫02πP′(eiθ)Q′(eiθ) dν(θ), z=eiθ, λ>0,where ν is the normalized Lebesgue measure and μ is a rational modification of ν. In this situation we analyse the algebraic results and the asymptotic behaviour of such orthogonal polynomials. Moreover some properties about the distribution of their zeros are given.