Zeros of Sobolev orthogonal polynomials on the unit circle (Q715130)

The authors consider the discrete Sobolev inner product associated with a nontrivial probability measure \(\mu\) supported on the unit circle \[ (f,g)_{S}=\int _{T} f(z)\overline {g(z)}d\mu (z)+\lambda f^{(j)}(a)\overline {g^{(j)}(a)},\tag{\(*\)} \] \(a\in \mathbb{C}\), \(\lambda\in \mathbb{R}^{+}-\{0\}\), \(j\in N\) and the orthonormal polynomials \(\{y_{n}\}_{n\geq 0}\) with respect to \((*)\). They obtain results regarding the behavior of the zeros of \(\{y_{n}\}_{n\geq 0}\) for a general \(j\) when \(n\) or \(\lambda\) tend to infinity. Also they analyze the behavior of the zeros according to the location of \(a\) by presenting some numerical computations of such zeros for the orthonormal polynomials associated with perturbations of the form \((*)\) for the Lebesgue and Bernstein-Szego measures on the unit circle.