On Some Recent Results on Asymptotic Behavior of Orthogonal Polynomials on the Unit Circle and Inserting Point Masses

In the present paper, we formulate and reprove, in a brief and self-contained presentation, some recent results concerning the asymptotic behavior of orthogonal polynomials on the unit circle by inserting point masses recently obtained by the authors and co-workers. In a first part, we deal with a spectral transformation of a Hermitian linear functional by the addition of the first derivative of a complex Dirac linear functional supported either in a point on the unit circle or in two symmetric points with respect to the unit circle. In this case, outer relative asymptotics for the new sequences of orthogonal polynomials in terms of the original ones are obtained. Necessary and sufficient conditions for the quasi-definiteness of the new linear functionals are given. The relation between the corresponding sequence of orthogonal polynomials in terms of the original one is presented. The second part is devoted to the study of a relevant family of orthogonal polynomials associated with perturbations of the original orthogonality measure by means of mass points: discrete Sobolev orthogonal polynomials. We compare the discrete Sobolev orthogonal polynomials with the initially ones. Finally, we analyze the behavior of their zeros.