Title: Differential properties for Sobolev orthogonality on the unit circle
Abstract: The aim of this paper is to study differential properties of the sequence of monic orthogonal polynomials with respect to the following Sobolev inner product:〈f,g〉s=∫02πf(eiθ)g(eiθ)dμ(θ)+1λ∫02πf′(eiθ)g′(eiθ)dθ2π,where μ is a finite positive Borel measure on [0,2π] verifying the following conditions: the Carathéodory function associated with μ has an analytic extension outside the unit disk and the induced norm is equivalent to the Lebesgue norm in the space L2. Here dθ/2π is the normalized Lebesgue measure and λ 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 Dirichlet boundary value problem is solved.