A family of Sobolev orthogonal polynomials on the unit circle
The authors consider the sequence of monic polynomials \(\Phi_n\) orthogonal with respect to the inner product \[ \langle f,g \rangle_s=\int_0^{2\pi} f(e^{i\theta}) \overline{g(e^{i\theta})} \frac{d \theta}{|e^{i \theta}-\alpha|^2} + \frac{1}{\lambda} \int_0^{2\pi} f'(e^{i\theta}) \overline{g'(e^{i\theta})} d \theta,, \] where \(\lambda >0\) and \(|\alpha|<1\). Algebraic expressions for \(\Phi_n\) and for their norms, and a three-term recurrence for \(\Phi_n\) are derived. From here, asymptotics for \(\Phi_n\) and the behavior of their zeros as \(n \to \infty\) is established. The paper is very much connected with another one by the same authors [\textit{E. Berriochoa} and \textit{A. Cachafeiro}, Rend. Mat. Appl., VII. Ser. 19, No. 1, 89-106 (1999; Zbl 0947.42018)].
scientific article; zbMATH DE number 1334859