Differential properties for Sobolev orthogonality on the unit circle

9 pages, no figures.-- MSC2000 code: 42C05.-- Issue title: Proceedings of the Fifth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Patras, 1999).MR#: MR1858282 (2002m:42022)Zbl#: Zbl 0990.42006The aim of this paper is to study differential properties of the sequence of monic orthogonal polynomials with respect to the following Sobolev inner product: $$\langle f, g\rangle_s= \int^{2\pi}_0 f(e^{i\theta}) \overline{g(e^{i\theta})} d\mu(\theta)+{1\over \lambda} \int^{2\pi}_0 f'(e^{i\theta}) \overline{g'(e^{i\theta})} {d\theta\over 2\pi},$$ where $\mu$ is a finite positive Borel measure on $[0, 2\pi]$ verifying the following conditions: the Carathéodory function associated with $\mu$ has an analytic extension outside the unit disk and the induced norm is equivalent to the Lebesgue norm in the space $L_2$. Here $d\theta/2\pi$ is the normalized Lebesgue measure and $\lambda$ is a positive real number. The nonhomogeneous second-order differential equations satisfied by the sequence of monic Sobolev orthogonal polynomials are obtained. Moreover, as an application, a sample of the Dirichlet boundary value problem is solved.The research was supported by DGES under grants number PB96-0344 and PB96-0120 C03-01.Publicad